What is the general solution for sec^4(x)=sec^2(x)tan^2(x)-2tan^4(x) ?

The general solution for sec^4(x)=sec^2(x)tan^2(x)-2tan^4(x) is No Solution for x\in\mathbb{R}

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sec^4(x)=sec^2(x)tan^2(x)-2tan^4(x)

Solution

sec^{4} (x) = sec^{2} (x) tan^{2} (x) - 2 tan^{4} (x)

Solution

No Solution for x \in R

Solution steps

sec^{4} (x) = sec^{2} (x) tan^{2} (x) - 2 tan^{4} (x)

Subtract

sec^{2} (x) tan^{2} (x) - 2 tan^{4} (x)

from both sides

sec^{4} (x) - sec^{2} (x) tan^{2} (x) + 2 tan^{4} (x) = 0

Express with sin, cos

sec^{4} (x) + 2 tan^{4} (x) - sec^{2} (x) tan^{2} (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= (\frac{1}{cos ( x )})^{4} + 2 tan^{4} (x) - (\frac{1}{cos ( x )})^{2} tan^{2} (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (\frac{1}{cos ( x )})^{4} + 2 (\frac{sin ( x )}{cos ( x )})^{4} - (\frac{1}{cos ( x )})^{2} (\frac{sin ( x )}{cos ( x )})^{2}

Simplify

(\frac{1}{cos ( x )})^{4} + 2 (\frac{sin ( x )}{cos ( x )})^{4} - (\frac{1}{cos ( x )})^{2} (\frac{sin ( x )}{cos ( x )})^{2} : \frac{1 + 2 sin ^{4} ( x ) - sin ^{2} ( x )}{cos ^{4} ( x )}

(\frac{1}{cos ( x )})^{4} + 2 (\frac{sin ( x )}{cos ( x )})^{4} - (\frac{1}{cos ( x )})^{2} (\frac{sin ( x )}{cos ( x )})^{2}

(\frac{1}{cos ( x )})^{4} = \frac{1}{cos ^{4} ( x )}

(\frac{1}{cos ( x )})^{4}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{4}}{cos ^{4} ( x )}

Apply rule

1^{a} = 1

1^{4} = 1

= \frac{1}{cos ^{4} ( x )}

2 (\frac{sin ( x )}{cos ( x )})^{4} = \frac{2 sin ^{4} ( x )}{cos ^{4} ( x )}

2 (\frac{sin ( x )}{cos ( x )})^{4}

(\frac{sin ( x )}{cos ( x )})^{4} = \frac{sin ^{4} ( x )}{cos ^{4} ( x )}

(\frac{sin ( x )}{cos ( x )})^{4}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{sin ^{4} ( x )}{cos ^{4} ( x )}

= 2 \cdot \frac{sin ^{4} ( x )}{cos ^{4} ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ^{4} ( x ) \cdot 2}{cos ^{4} ( x )}

(\frac{1}{cos ( x )})^{2} (\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{2} ( x )}{cos ^{4} ( x )}

(\frac{1}{cos ( x )})^{2} (\frac{sin ( x )}{cos ( x )})^{2}

(\frac{1}{cos ( x )})^{2} = \frac{1}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( x )}

= (\frac{sin ( x )}{cos ( x )})^{2} \frac{1}{cos ^{2} ( x )}

(\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

(\frac{sin ( x )}{cos ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{1}{cos ^{2} ( x )} \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot sin ^{2} ( x )}{cos ^{2} ( x ) cos ^{2} ( x )}

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ^{2} ( x ) cos ^{2} ( x )}

cos^{2} (x) cos^{2} (x) = cos^{4} (x)

cos^{2} (x) cos^{2} (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos^{2} (x) = cos^{2 + 2} (x)

= cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= cos^{4} (x)

= \frac{sin ^{2} ( x )}{cos ^{4} ( x )}

= \frac{1}{cos ^{4} ( x )} + \frac{2 sin ^{4} ( x )}{cos ^{4} ( x )} - \frac{sin ^{2} ( x )}{cos ^{4} ( x )}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 2 sin ^{4} ( x ) - sin ^{2} ( x )}{cos ^{4} ( x )}

= \frac{1 + 2 sin ^{4} ( x ) - sin ^{2} ( x )}{cos ^{4} ( x )}

\frac{1 - sin ^{2} ( x ) + 2 sin ^{4} ( x )}{cos ^{4} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 - sin^{2} (x) + 2 sin^{4} (x) = 0

Solve by substitution

1 - sin^{2} (x) + 2 sin^{4} (x) = 0

Let:

sin (x) = u

1 - u^{2} + 2 u^{4} = 0

1 - u^{2} + 2 u^{4} = 0 : u = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

1 - u^{2} + 2 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

2 u^{4} - u^{2} + 1 = 0

Rewrite the equation with

a = u^{2}

and

a^{2} = u^{4}

2 a^{2} - a + 1 = 0

Solve

2 a^{2} - a + 1 = 0 : a = \frac{1}{4} + i \frac{7}{4}, a = \frac{1}{4} - i \frac{7}{4}

2 a^{2} - a + 1 = 0

Solve with the quadratic formula

2 a^{2} - a + 1 = 0

Quadratic Equation Formula:

For

a = 2, b = - 1, c = 1

a_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

a_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

Simplify

(- 1)^{2} - 4 \cdot 2 \cdot 1 : 7 i

(- 1)^{2} - 4 \cdot 2 \cdot 1

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 8

= 1 - 8

Subtract the numbers:

1 - 8 = - 7

= - 7

Apply radical rule:

- a = - 1 a

- 7 = - 1 7

= - 1 7

Apply imaginary number rule:

- 1 = i

= 7 i

a_{1, 2} = \frac{- ( - 1 ) \pm 7 i}{2 \cdot 2}

Separate the solutions

a_{1} = \frac{- ( - 1 ) + 7 i}{2 \cdot 2}, a_{2} = \frac{- ( - 1 ) - 7 i}{2 \cdot 2}

a = \frac{- ( - 1 ) + 7 i}{2 \cdot 2} : \frac{1}{4} + i \frac{7}{4}

\frac{- ( - 1 ) + 7 i}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 + 7 i}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1 + 7 i}{4}

Rewrite

\frac{1 + 7 i}{4}

in standard complex form:

\frac{1}{4} + \frac{7}{4} i

\frac{1 + 7 i}{4}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + 7 i}{4} = \frac{1}{4} + \frac{7 i}{4}

= \frac{1}{4} + \frac{7 i}{4}

= \frac{1}{4} + \frac{7}{4} i

a = \frac{- ( - 1 ) - 7 i}{2 \cdot 2} : \frac{1}{4} - i \frac{7}{4}

\frac{- ( - 1 ) - 7 i}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 - 7 i}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1 - 7 i}{4}

Rewrite

\frac{1 - 7 i}{4}

in standard complex form:

\frac{1}{4} - \frac{7}{4} i

\frac{1 - 7 i}{4}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 - 7 i}{4} = \frac{1}{4} - \frac{7 i}{4}

= \frac{1}{4} - \frac{7 i}{4}

= \frac{1}{4} - \frac{7}{4} i

The solutions to the quadratic equation are:

a = \frac{1}{4} + i \frac{7}{4}, a = \frac{1}{4} - i \frac{7}{4}

a = \frac{1}{4} + i \frac{7}{4}, a = \frac{1}{4} - i \frac{7}{4}

Substitute back

a = u^{2},

solve for

u

Solve

u^{2} = \frac{1}{4} + i \frac{7}{4} : u = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

u^{2} = \frac{1}{4} + i \frac{7}{4}

Substitute

u = a + bi

(a + bi)^{2} = \frac{1}{4} + i \frac{7}{4}

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = \frac{1}{4} + i \frac{7}{4}

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = \frac{1}{4} 2 ab = \frac{7}{4}]

[a^{2} - b^{2} = \frac{1}{4} 2 ab = \frac{7}{4}] : a = \frac{7}{2 2 2 2 - 1}, a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} b = - \frac{2 2 - 1}{2 2}

[a^{2} - b^{2} = \frac{1}{4} 2 ab = \frac{7}{4}]

Isolate

a

for

2 ab = \frac{7}{4} : a = \frac{7}{8 b}

2 ab = \frac{7}{4}

Divide both sides by

2 b

2 ab = \frac{7}{4}

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{\frac{7}{4}}{2 b}

Simplify

\frac{2 ab}{2 b} = \frac{\frac{7}{4}}{2 b}

Simplify

\frac{2 ab}{2 b} : a

\frac{2 ab}{2 b}

Divide the numbers:

\frac{2}{2} = 1

= \frac{ab}{b}

Cancel the common factor:

b

= a

Simplify

\frac{\frac{7}{4}}{2 b} : \frac{7}{8 b}

\frac{\frac{7}{4}}{2 b}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{7}{4 \cdot 2 b}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{7}{8 b}

a = \frac{7}{8 b}

a = \frac{7}{8 b}

a = \frac{7}{8 b}

Plug the solutions

a = \frac{7}{8 b}

into

a^{2} - b^{2} = \frac{1}{4}

For

a^{2} - b^{2} = \frac{1}{4}

, subsitute

a

with

\frac{7}{8 b} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

For

a^{2} - b^{2} = \frac{1}{4}

, subsitute

a

with

\frac{7}{8 b}

(\frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Solve

(\frac{7}{8 b})^{2} - b^{2} = \frac{1}{4} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

(\frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Multiply by LCM

(\frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Simplify

(\frac{7}{8 b})^{2} : \frac{7}{64 b ^{2}}

(\frac{7}{8 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 7 ) ^{2}}{( 8 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(8 b)^{2} = 8^{2} b^{2}

= \frac{( 7 ) ^{2}}{8 ^{2} b ^{2}}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= \frac{7}{8 ^{2} b ^{2}}

8^{2} = 64

= \frac{7}{64 b ^{2}}

\frac{7}{64 b ^{2}} - b^{2} = \frac{1}{4}

Find Least Common Multiplier of

64 b^{2}, 4 : 64 b^{2}

64 b^{2}, 4

Lowest Common Multiplier (LCM)

Least Common Multiplier of

64, 4 : 64

64, 4

Least Common Multiplier (LCM)

Prime factorization of

64 : 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

64

4

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64

= 64

Compute an expression comprised of factors that appear either in

64 b^{2}

4

= 64 b^{2}

Multiply by LCM=

64 b^{2}

\frac{7}{64 b ^{2}} \cdot 64 b^{2} - b^{2} \cdot 64 b^{2} = \frac{1}{4} \cdot 64 b^{2}

Simplify

\frac{7}{64 b ^{2}} \cdot 64 b^{2} - b^{2} \cdot 64 b^{2} = \frac{1}{4} \cdot 64 b^{2}

Simplify

\frac{7}{64 b ^{2}} \cdot 64 b^{2} : 7

\frac{7}{64 b ^{2}} \cdot 64 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 \cdot 64 b ^{2}}{64 b ^{2}}

Cancel the common factor:

64

= \frac{7 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 7

Simplify

- b^{2} \cdot 64 b^{2} : - 64 b^{4}

- b^{2} \cdot 64 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 64 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 64 b^{4}

Simplify

\frac{1}{4} \cdot 64 b^{2} : 16 b^{2}

\frac{1}{4} \cdot 64 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 64}{4} b^{2}

\frac{1 \cdot 64}{4} = 16

\frac{1 \cdot 64}{4}

Multiply the numbers:

1 \cdot 64 = 64

= \frac{64}{4}

Divide the numbers:

\frac{64}{4} = 16

= 16

= 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

Solve

7 - 64 b^{4} = 16 b^{2} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

7 - 64 b^{4} = 16 b^{2}

Move

16 b^{2}

to the left side

7 - 64 b^{4} = 16 b^{2}

Subtract

16 b^{2}

from both sides

7 - 64 b^{4} - 16 b^{2} = 16 b^{2} - 16 b^{2}

Simplify

7 - 64 b^{4} - 16 b^{2} = 0

7 - 64 b^{4} - 16 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 64 b^{4} - 16 b^{2} + 7 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 64 u^{2} - 16 u + 7 = 0

Solve

- 64 u^{2} - 16 u + 7 = 0 : u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

- 64 u^{2} - 16 u + 7 = 0

Solve with the quadratic formula

- 64 u^{2} - 16 u + 7 = 0

Quadratic Equation Formula:

For

a = - 64, b = - 16, c = 7

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 ( - 64 ) \cdot 7}{2 ( - 64 )}

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 ( - 64 ) \cdot 7}{2 ( - 64 )}

(- 16)^{2} - 4 (- 64) \cdot 7 = 322

(- 16)^{2} - 4 (- 64) \cdot 7

Apply rule

- (- a) = a

= (- 16)^{2} + 4 \cdot 64 \cdot 7

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 16)^{2} = 1 6^{2}

= 1 6^{2} + 4 \cdot 64 \cdot 7

Multiply the numbers:

4 \cdot 64 \cdot 7 = 1792

= 1 6^{2} + 1792

1 6^{2} = 256

= 256 + 1792

Add the numbers:

256 + 1792 = 2048

= 2048

Prime factorization of

2048 : 2^{11}

2048

2048

divides by

22048 = 1024 \cdot 2

= 2 \cdot 1024

1024

divides by

21024 = 512 \cdot 2

= 2 \cdot 2 \cdot 512

512

divides by

2512 = 256 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 256

256

divides by

2256 = 128 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 128

128

divides by

2128 = 64 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{11}

= 2^{11}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{10} \cdot 2

Apply radical rule:

= 2 2^{10}

Apply radical rule:

2^{10} = 2^{\frac{10}{2}} = 2^{5}

= 2^{5} 2

Refine

= 322

u_{1, 2} = \frac{- ( - 16 ) \pm 32 2}{2 ( - 64 )}

Separate the solutions

u_{1} = \frac{- ( - 16 ) + 32 2}{2 ( - 64 )}, u_{2} = \frac{- ( - 16 ) - 32 2}{2 ( - 64 )}

u = \frac{- ( - 16 ) + 32 2}{2 ( - 64 )} : - \frac{1 + 2 2}{8}

\frac{- ( - 16 ) + 32 2}{2 ( - 64 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{16 + 32 2}{- 2 \cdot 64}

Multiply the numbers:

2 \cdot 64 = 128

= \frac{16 + 32 2}{- 128}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{16 + 32 2}{128}

Cancel

\frac{16 + 32 2}{128} : \frac{1 + 2 2}{8}

\frac{16 + 32 2}{128}

Factor

16 + 322 : 16 (1 + 22)

16 + 322

Rewrite as

= 16 \cdot 1 + 16 \cdot 22

Factor out common term

16

= 16 (1 + 22)

= \frac{16 ( 1 + 2 2 )}{128}

Cancel the common factor:

16

= \frac{1 + 2 2}{8}

= - \frac{1 + 2 2}{8}

u = \frac{- ( - 16 ) - 32 2}{2 ( - 64 )} : \frac{2 2 - 1}{8}

\frac{- ( - 16 ) - 32 2}{2 ( - 64 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{16 - 32 2}{- 2 \cdot 64}

Multiply the numbers:

2 \cdot 64 = 128

= \frac{16 - 32 2}{- 128}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

16 - 322 = - (322 - 16)

= \frac{32 2 - 16}{128}

Factor

322 - 16 : 16 (22 - 1)

322 - 16

Rewrite as

= 16 \cdot 22 - 16 \cdot 1

Factor out common term

16

= 16 (22 - 1)

= \frac{16 ( 2 2 - 1 )}{128}

Cancel the common factor:

16

= \frac{2 2 - 1}{8}

The solutions to the quadratic equation are:

u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{1 + 2 2}{8} :

No Solution for

b \in R

b^{2} = - \frac{1 + 2 2}{8}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 2 - 1}{8} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

b^{2} = \frac{2 2 - 1}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 2 - 1}{8}, b = - \frac{2 2 - 1}{8}

\frac{2 2 - 1}{8} = \frac{2 2 - 1}{2 2}

\frac{2 2 - 1}{8}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{2 2 - 1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

2^{3} = 2^{2} \cdot 2

= 2^{2} \cdot 2

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 = 2^{2} 2

= 2^{2} 2

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 22

= \frac{2 2 - 1}{2 2}

- \frac{2 2 - 1}{8} = - \frac{2 2 - 1}{2 2}

- \frac{2 2 - 1}{8}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{2 2 - 1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

2^{3} = 2^{2} \cdot 2

= 2^{2} \cdot 2

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 = 2^{2} 2

= 2^{2} 2

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 22

= - \frac{2 2 - 1}{2 2}

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

The solutions are

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(\frac{7}{8 b})^{2} - b^{2}

and compare to zero

Solve

8 b = 0 : b = 0

8 b = 0

Divide both sides by

8

8 b = 0

Divide both sides by

8

\frac{8 b}{8} = \frac{0}{8}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

Plug the solutions

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

into

2 ab = \frac{7}{4}

For

2 ab = \frac{7}{4}

, subsitute

b

with

\frac{2 2 - 1}{2 2} : a = \frac{7}{2 2 2 2 - 1}

For

2 ab = \frac{7}{4}

, subsitute

b

with

\frac{2 2 - 1}{2 2}

2 a \frac{2 2 - 1}{2 2} = \frac{7}{4}

Solve

2 a \frac{2 2 - 1}{2 2} = \frac{7}{4} : a = \frac{7}{2 2 2 2 - 1}

2 a \frac{2 2 - 1}{2 2} = \frac{7}{4}

Multiply both sides by

22

2 a \frac{2 2 - 1}{2 2} = \frac{7}{4}

Multiply both sides by

22

2 a \frac{2 2 - 1}{2 2} \cdot 22 = \frac{7 \cdot 2 2}{4}

Simplify

2 a \frac{2 2 - 1}{2 2} \cdot 22 = \frac{7 \cdot 2 2}{4}

Simplify

2 a \frac{2 2 - 1}{2 2} \cdot 22 : 2 a 22 - 1

2 a \frac{2 2 - 1}{2 2} \cdot 22

2 \cdot 2 = 2^{2}

2 \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2 = 2^{1 + 1}

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} a \frac{2 2 - 1}{2 2} 2

Apply radical rule:

a = a^{\frac{1}{2}}

2 = 2^{\frac{1}{2}}

= 2^{2} a \frac{2 2 - 1}{2 2} \cdot 2^{\frac{1}{2}}

2^{2} \cdot 2^{\frac{1}{2}} = 2^{\frac{5}{2}}

2^{2} \cdot 2^{\frac{1}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{2} \cdot 2^{\frac{1}{2}} = 2^{2 + \frac{1}{2}}

= 2^{2 + \frac{1}{2}}

2 + \frac{1}{2} = \frac{5}{2}

2 + \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= \frac{2 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 \cdot 2 + 1}{2}

2 \cdot 2 + 1 = 5

2 \cdot 2 + 1

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 1

Add the numbers:

4 + 1 = 5

= 5

= \frac{5}{2}

= 2^{\frac{5}{2}}

= 2^{\frac{5}{2}} a \frac{2 2 - 1}{2 2}

Apply the fraction rule:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2}

Cancel

\frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2} : 2 a 22 - 1

\frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2}

\frac{2 ^{\frac{5}{2}}}{2 2} = 2

\frac{2 ^{\frac{5}{2}}}{2 2}

Simplify

\frac{2 ^{\frac{5}{2}}}{2} : 2^{\frac{3}{2}}

\frac{2 ^{\frac{5}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

= 2^{\frac{5}{2} - 1}

\frac{5}{2} - 1 = \frac{3}{2}

\frac{5}{2} - 1

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= - \frac{1 \cdot 2}{2} + \frac{5}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 \cdot 2 + 5}{2}

- 1 \cdot 2 + 5 = 3

- 1 \cdot 2 + 5

Multiply the numbers:

1 \cdot 2 = 2

= - 2 + 5

Add/Subtract the numbers:

- 2 + 5 = 3

= 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

= \frac{2 ^{\frac{3}{2}}}{2}

Apply radical rule:

a = a^{\frac{1}{2}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}}

Simplify

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}} : 2

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

= 2^{\frac{3}{2} - \frac{1}{2}}

\frac{3}{2} - \frac{1}{2} = 1

\frac{3}{2} - \frac{1}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 - 1}{2}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply exponent rule:

a^{1} = a

= 2

= 2

= 2 a 22 - 1

= 2 a 22 - 1

Simplify

\frac{7 \cdot 2 2}{4} : \frac{7}{2}

\frac{7 \cdot 2 2}{4}

Factor the number:

4 = 2 \cdot 2

= \frac{7 \cdot 2 2}{2 \cdot 2}

Cancel the common factor:

2

= \frac{7 2}{2}

Apply radical rule:

a = a a

2 = 22

= \frac{7 2}{2 2}

Cancel the common factor:

2

= \frac{7}{2}

2 a 22 - 1 = \frac{7}{2}

2 a 22 - 1 = \frac{7}{2}

2 a 22 - 1 = \frac{7}{2}

Divide both sides by

2 22 - 1

2 a 22 - 1 = \frac{7}{2}

Divide both sides by

2 22 - 1

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{\frac{7}{2}}{2 2 2 - 1}

Simplify

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{\frac{7}{2}}{2 2 2 - 1}

Simplify

\frac{2 a 2 2 - 1}{2 2 2 - 1} : a

\frac{2 a 2 2 - 1}{2 2 2 - 1}

Cancel the common factor:

2

= \frac{a 2 2 - 1}{2 2 - 1}

Cancel the common factor:

22 - 1

= a

Simplify

\frac{\frac{7}{2}}{2 2 2 - 1} : \frac{7}{2 2 2 2 - 1}

\frac{\frac{7}{2}}{2 2 2 - 1}

Apply the fraction rule:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

= \frac{7}{2 \cdot 2 2 2 - 1}

= \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

For

2 ab = \frac{7}{4}

, subsitute

b

with

- \frac{2 2 - 1}{2 2} : a = - \frac{7}{2 2 2 2 - 1}

For

2 ab = \frac{7}{4}

, subsitute

b

with

- \frac{2 2 - 1}{2 2}

2 a (- \frac{2 2 - 1}{2 2}) = \frac{7}{4}

Solve

2 a (- \frac{2 2 - 1}{2 2}) = \frac{7}{4} : a = - \frac{7}{2 2 2 2 - 1}

2 a (- \frac{2 2 - 1}{2 2}) = \frac{7}{4}

Divide both sides by

2 (- \frac{2 2 - 1}{2 2})

2 a (- \frac{2 2 - 1}{2 2}) = \frac{7}{4}

Divide both sides by

2 (- \frac{2 2 - 1}{2 2})

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} = \frac{\frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} = \frac{\frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} : a

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} : \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

2 a (- \frac{2 2 - 1}{2 2}) = - 2 a \frac{2 2 - 1}{2 2}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2 2}) = - 2 \cdot \frac{2 2 - 1}{2 2}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

Cancel the common factor:

- 2

= \frac{a \frac{2 2 - 1}{2 2}}{\frac{2 2 - 1}{2 2}}

Cancel the common factor:

\frac{2 2 - 1}{2 2}

= a

Simplify

\frac{\frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )} : - \frac{7}{2 2 2 2 - 1}

\frac{\frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply the fraction rule:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

= \frac{7}{4 \cdot 2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

4 \cdot 2 (- \frac{2 2 - 1}{2 2}) = - 4 \cdot 2 \cdot \frac{2 2 - 1}{2 2}

= \frac{7}{- 4 \cdot 2 \cdot \frac{2 2 - 1}{2 2}}

- 4 \cdot 2 \cdot \frac{2 2 - 1}{2 2} = - 2 \cdot 2 22 - 1

- 4 \cdot 2 \cdot \frac{2 2 - 1}{2 2}

Convert

4

to fraction

: \frac{4}{1}

4

Convert element to fraction:

4 = \frac{4}{1}

= \frac{4}{1}

= - \frac{4}{1} \cdot 2 \cdot \frac{2 2 - 1}{2 2}

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

= - \frac{4}{1} \cdot \frac{2}{1} \cdot \frac{2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

\frac{4}{1} \cdot \frac{2}{1} \cdot \frac{2 2 - 1}{2 2} = \frac{4 \cdot 2 2 2 - 1}{1 \cdot 1 \cdot 2 2}

= - \frac{4 \cdot 2 2 2 - 1}{1 \cdot 1 \cdot 2 2}

\frac{4 \cdot 2 2 2 - 1}{1 \cdot 1 \cdot 2 2} = 2 \cdot 2 22 - 1

\frac{4 \cdot 2 2 2 - 1}{1 \cdot 1 \cdot 2 2}

Multiply the numbers:

1 \cdot 1 \cdot 2 = 2

= \frac{4 \cdot 2 2 2 - 1}{2 2}

Cancel the common factor:

2

= \frac{4 2 2 - 1}{2}

Factor the number:

4 = 2 \cdot 2

= \frac{2 \cdot 2 2 2 - 1}{2}

Apply radical rule:

a = a a

2 = 22

= \frac{2 2 \cdot 2 2 2 - 1}{2}

Cancel the common factor:

2

= 2 \cdot 2 22 - 1

= - 2 \cdot 2 22 - 1

= \frac{7}{- 2 \cdot 2 2 2 - 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{7}{2 \cdot 2 2 2 - 1}

= - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = \frac{1}{4}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2} :

True

a^{2} - b^{2} = \frac{1}{4}

Plug in

a = - \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2}

(- \frac{7}{2 2 2 2 - 1})^{2} - (- \frac{2 2 - 1}{2 2})^{2} = \frac{1}{4}

Refine

\frac{1}{4} = \frac{1}{4}

True

Check the solution

a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} :

True

a^{2} - b^{2} = \frac{1}{4}

Plug in

a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2}

(\frac{7}{2 2 2 2 - 1})^{2} - (\frac{2 2 - 1}{2 2})^{2} = \frac{1}{4}

Refine

\frac{1}{4} = \frac{1}{4}

True

Check the solutions by plugging them into

2 ab = \frac{7}{4}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2} :

True

2 ab = \frac{7}{4}

Plug in

a = - \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2}

2 (- \frac{7}{2 2 2 2 - 1}) (- \frac{2 2 - 1}{2 2}) = \frac{7}{4}

Refine

\frac{7}{4} = \frac{7}{4}

True

Check the solution

a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} :

True

2 ab = \frac{7}{4}

Plug in

a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2}

2 \cdot \frac{7}{2 2 2 2 - 1} \cdot \frac{2 2 - 1}{2 2} = \frac{7}{4}

Refine

\frac{7}{4} = \frac{7}{4}

True

Therefore, the final solutions for

a^{2} - b^{2} = \frac{1}{4}, 2 ab = \frac{7}{4}

are

a = \frac{7}{2 2 2 2 - 1}, a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} b = - \frac{2 2 - 1}{2 2}

Substitute back

u = a + bi

u = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

Solve

u^{2} = \frac{1}{4} - i \frac{7}{4} : u = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

u^{2} = \frac{1}{4} - i \frac{7}{4}

Substitute

u = a + bi

(a + bi)^{2} = \frac{1}{4} - i \frac{7}{4}

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = \frac{1}{4} - i \frac{7}{4}

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = \frac{1}{4} 2 ab = - \frac{7}{4}]

[a^{2} - b^{2} = \frac{1}{4} 2 ab = - \frac{7}{4}] : a = - \frac{7}{2 2 2 2 - 1}, a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} b = - \frac{2 2 - 1}{2 2}

[a^{2} - b^{2} = \frac{1}{4} 2 ab = - \frac{7}{4}]

Isolate

a

for

2 ab = - \frac{7}{4} : a = - \frac{7}{8 b}

2 ab = - \frac{7}{4}

Divide both sides by

2 b

2 ab = - \frac{7}{4}

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{- \frac{7}{4}}{2 b}

Simplify

\frac{2 ab}{2 b} = \frac{- \frac{7}{4}}{2 b}

Simplify

\frac{2 ab}{2 b} : a

\frac{2 ab}{2 b}

Divide the numbers:

\frac{2}{2} = 1

= \frac{ab}{b}

Cancel the common factor:

b

= a

Simplify

\frac{- \frac{7}{4}}{2 b} : - \frac{7}{8 b}

\frac{- \frac{7}{4}}{2 b}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{7}{4}}{2 b}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{7}{4}}{2 b} = \frac{7}{4 \cdot 2 b}

= - \frac{7}{4 \cdot 2 b}

Multiply the numbers:

4 \cdot 2 = 8

= - \frac{7}{8 b}

a = - \frac{7}{8 b}

a = - \frac{7}{8 b}

a = - \frac{7}{8 b}

Plug the solutions

a = - \frac{7}{8 b}

into

a^{2} - b^{2} = \frac{1}{4}

For

a^{2} - b^{2} = \frac{1}{4}

, subsitute

a

with

- \frac{7}{8 b} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

For

a^{2} - b^{2} = \frac{1}{4}

, subsitute

a

with

- \frac{7}{8 b}

(- \frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Solve

(- \frac{7}{8 b})^{2} - b^{2} = \frac{1}{4} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

(- \frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Multiply by LCM

(- \frac{7}{8 b})^{2} - b^{2} = \frac{1}{4}

Simplify

(- \frac{7}{8 b})^{2} : \frac{7}{64 b ^{2}}

(- \frac{7}{8 b})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{7}{8 b})^{2} = (\frac{7}{8 b})^{2}

= (\frac{7}{8 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 7 ) ^{2}}{( 8 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(8 b)^{2} = 8^{2} b^{2}

= \frac{( 7 ) ^{2}}{8 ^{2} b ^{2}}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= \frac{7}{8 ^{2} b ^{2}}

8^{2} = 64

= \frac{7}{64 b ^{2}}

\frac{7}{64 b ^{2}} - b^{2} = \frac{1}{4}

Find Least Common Multiplier of

64 b^{2}, 4 : 64 b^{2}

64 b^{2}, 4

Lowest Common Multiplier (LCM)

Least Common Multiplier of

64, 4 : 64

64, 4

Least Common Multiplier (LCM)

Prime factorization of

64 : 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

64

4

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64

= 64

Compute an expression comprised of factors that appear either in

64 b^{2}

4

= 64 b^{2}

Multiply by LCM=

64 b^{2}

\frac{7}{64 b ^{2}} \cdot 64 b^{2} - b^{2} \cdot 64 b^{2} = \frac{1}{4} \cdot 64 b^{2}

Simplify

\frac{7}{64 b ^{2}} \cdot 64 b^{2} - b^{2} \cdot 64 b^{2} = \frac{1}{4} \cdot 64 b^{2}

Simplify

\frac{7}{64 b ^{2}} \cdot 64 b^{2} : 7

\frac{7}{64 b ^{2}} \cdot 64 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 \cdot 64 b ^{2}}{64 b ^{2}}

Cancel the common factor:

64

= \frac{7 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 7

Simplify

- b^{2} \cdot 64 b^{2} : - 64 b^{4}

- b^{2} \cdot 64 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 64 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 64 b^{4}

Simplify

\frac{1}{4} \cdot 64 b^{2} : 16 b^{2}

\frac{1}{4} \cdot 64 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 64}{4} b^{2}

\frac{1 \cdot 64}{4} = 16

\frac{1 \cdot 64}{4}

Multiply the numbers:

1 \cdot 64 = 64

= \frac{64}{4}

Divide the numbers:

\frac{64}{4} = 16

= 16

= 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

7 - 64 b^{4} = 16 b^{2}

Solve

7 - 64 b^{4} = 16 b^{2} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

7 - 64 b^{4} = 16 b^{2}

Move

16 b^{2}

to the left side

7 - 64 b^{4} = 16 b^{2}

Subtract

16 b^{2}

from both sides

7 - 64 b^{4} - 16 b^{2} = 16 b^{2} - 16 b^{2}

Simplify

7 - 64 b^{4} - 16 b^{2} = 0

7 - 64 b^{4} - 16 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 64 b^{4} - 16 b^{2} + 7 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 64 u^{2} - 16 u + 7 = 0

Solve

- 64 u^{2} - 16 u + 7 = 0 : u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

- 64 u^{2} - 16 u + 7 = 0

Solve with the quadratic formula

- 64 u^{2} - 16 u + 7 = 0

Quadratic Equation Formula:

For

a = - 64, b = - 16, c = 7

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 ( - 64 ) \cdot 7}{2 ( - 64 )}

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 ( - 64 ) \cdot 7}{2 ( - 64 )}

(- 16)^{2} - 4 (- 64) \cdot 7 = 322

(- 16)^{2} - 4 (- 64) \cdot 7

Apply rule

- (- a) = a

= (- 16)^{2} + 4 \cdot 64 \cdot 7

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 16)^{2} = 1 6^{2}

= 1 6^{2} + 4 \cdot 64 \cdot 7

Multiply the numbers:

4 \cdot 64 \cdot 7 = 1792

= 1 6^{2} + 1792

1 6^{2} = 256

= 256 + 1792

Add the numbers:

256 + 1792 = 2048

= 2048

Prime factorization of

2048 : 2^{11}

2048

2048

divides by

22048 = 1024 \cdot 2

= 2 \cdot 1024

1024

divides by

21024 = 512 \cdot 2

= 2 \cdot 2 \cdot 512

512

divides by

2512 = 256 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 256

256

divides by

2256 = 128 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 128

128

divides by

2128 = 64 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{11}

= 2^{11}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{10} \cdot 2

Apply radical rule:

= 2 2^{10}

Apply radical rule:

2^{10} = 2^{\frac{10}{2}} = 2^{5}

= 2^{5} 2

Refine

= 322

u_{1, 2} = \frac{- ( - 16 ) \pm 32 2}{2 ( - 64 )}

Separate the solutions

u_{1} = \frac{- ( - 16 ) + 32 2}{2 ( - 64 )}, u_{2} = \frac{- ( - 16 ) - 32 2}{2 ( - 64 )}

u = \frac{- ( - 16 ) + 32 2}{2 ( - 64 )} : - \frac{1 + 2 2}{8}

\frac{- ( - 16 ) + 32 2}{2 ( - 64 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{16 + 32 2}{- 2 \cdot 64}

Multiply the numbers:

2 \cdot 64 = 128

= \frac{16 + 32 2}{- 128}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{16 + 32 2}{128}

Cancel

\frac{16 + 32 2}{128} : \frac{1 + 2 2}{8}

\frac{16 + 32 2}{128}

Factor

16 + 322 : 16 (1 + 22)

16 + 322

Rewrite as

= 16 \cdot 1 + 16 \cdot 22

Factor out common term

16

= 16 (1 + 22)

= \frac{16 ( 1 + 2 2 )}{128}

Cancel the common factor:

16

= \frac{1 + 2 2}{8}

= - \frac{1 + 2 2}{8}

u = \frac{- ( - 16 ) - 32 2}{2 ( - 64 )} : \frac{2 2 - 1}{8}

\frac{- ( - 16 ) - 32 2}{2 ( - 64 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{16 - 32 2}{- 2 \cdot 64}

Multiply the numbers:

2 \cdot 64 = 128

= \frac{16 - 32 2}{- 128}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

16 - 322 = - (322 - 16)

= \frac{32 2 - 16}{128}

Factor

322 - 16 : 16 (22 - 1)

322 - 16

Rewrite as

= 16 \cdot 22 - 16 \cdot 1

Factor out common term

16

= 16 (22 - 1)

= \frac{16 ( 2 2 - 1 )}{128}

Cancel the common factor:

16

= \frac{2 2 - 1}{8}

The solutions to the quadratic equation are:

u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

u = - \frac{1 + 2 2}{8}, u = \frac{2 2 - 1}{8}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{1 + 2 2}{8} :

No Solution for

b \in R

b^{2} = - \frac{1 + 2 2}{8}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 2 - 1}{8} : b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

b^{2} = \frac{2 2 - 1}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 2 - 1}{8}, b = - \frac{2 2 - 1}{8}

\frac{2 2 - 1}{8} = \frac{2 2 - 1}{2 2}

\frac{2 2 - 1}{8}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{2 2 - 1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

2^{3} = 2^{2} \cdot 2

= 2^{2} \cdot 2

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 = 2^{2} 2

= 2^{2} 2

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 22

= \frac{2 2 - 1}{2 2}

- \frac{2 2 - 1}{8} = - \frac{2 2 - 1}{2 2}

- \frac{2 2 - 1}{8}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{2 2 - 1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

2^{3} = 2^{2} \cdot 2

= 2^{2} \cdot 2

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 = 2^{2} 2

= 2^{2} 2

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 22

= - \frac{2 2 - 1}{2 2}

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

The solutions are

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(- \frac{7}{8 b})^{2} - b^{2}

and compare to zero

Solve

8 b = 0 : b = 0

8 b = 0

Divide both sides by

8

8 b = 0

Divide both sides by

8

\frac{8 b}{8} = \frac{0}{8}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

Plug the solutions

b = \frac{2 2 - 1}{2 2}, b = - \frac{2 2 - 1}{2 2}

into

2 ab = - \frac{7}{4}

For

2 ab = - \frac{7}{4}

, subsitute

b

with

\frac{2 2 - 1}{2 2} : a = - \frac{7}{2 2 2 2 - 1}

For

2 ab = - \frac{7}{4}

, subsitute

b

with

\frac{2 2 - 1}{2 2}

2 a \frac{2 2 - 1}{2 2} = - \frac{7}{4}

Solve

2 a \frac{2 2 - 1}{2 2} = - \frac{7}{4} : a = - \frac{7}{2 2 2 2 - 1}

2 a \frac{2 2 - 1}{2 2} = - \frac{7}{4}

Multiply both sides by

22

2 a \frac{2 2 - 1}{2 2} = - \frac{7}{4}

Multiply both sides by

22

2 a \frac{2 2 - 1}{2 2} \cdot 22 = (- \frac{7}{4}) \cdot 22

Simplify

2 a \frac{2 2 - 1}{2 2} \cdot 22 = (- \frac{7}{4}) \cdot 22

Simplify

2 a \frac{2 2 - 1}{2 2} \cdot 22 : 2 a 22 - 1

2 a \frac{2 2 - 1}{2 2} \cdot 22

2 \cdot 2 = 2^{2}

2 \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2 = 2^{1 + 1}

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} a \frac{2 2 - 1}{2 2} 2

Apply radical rule:

a = a^{\frac{1}{2}}

2 = 2^{\frac{1}{2}}

= 2^{2} a \frac{2 2 - 1}{2 2} \cdot 2^{\frac{1}{2}}

2^{2} \cdot 2^{\frac{1}{2}} = 2^{\frac{5}{2}}

2^{2} \cdot 2^{\frac{1}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{2} \cdot 2^{\frac{1}{2}} = 2^{2 + \frac{1}{2}}

= 2^{2 + \frac{1}{2}}

2 + \frac{1}{2} = \frac{5}{2}

2 + \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= \frac{2 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 \cdot 2 + 1}{2}

2 \cdot 2 + 1 = 5

2 \cdot 2 + 1

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 1

Add the numbers:

4 + 1 = 5

= 5

= \frac{5}{2}

= 2^{\frac{5}{2}}

= 2^{\frac{5}{2}} a \frac{2 2 - 1}{2 2}

Apply the fraction rule:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2}

Cancel

\frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2} : 2 a 22 - 1

\frac{2 ^{\frac{5}{2}} a 2 2 - 1}{2 2}

\frac{2 ^{\frac{5}{2}}}{2 2} = 2

\frac{2 ^{\frac{5}{2}}}{2 2}

Simplify

\frac{2 ^{\frac{5}{2}}}{2} : 2^{\frac{3}{2}}

\frac{2 ^{\frac{5}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

= 2^{\frac{5}{2} - 1}

\frac{5}{2} - 1 = \frac{3}{2}

\frac{5}{2} - 1

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= - \frac{1 \cdot 2}{2} + \frac{5}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 \cdot 2 + 5}{2}

- 1 \cdot 2 + 5 = 3

- 1 \cdot 2 + 5

Multiply the numbers:

1 \cdot 2 = 2

= - 2 + 5

Add/Subtract the numbers:

- 2 + 5 = 3

= 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

= \frac{2 ^{\frac{3}{2}}}{2}

Apply radical rule:

a = a^{\frac{1}{2}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}}

Simplify

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}} : 2

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

= 2^{\frac{3}{2} - \frac{1}{2}}

\frac{3}{2} - \frac{1}{2} = 1

\frac{3}{2} - \frac{1}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 - 1}{2}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply exponent rule:

a^{1} = a

= 2

= 2

= 2 a 22 - 1

= 2 a 22 - 1

Simplify

(- \frac{7}{4}) \cdot 22 : - \frac{7}{2} 2

(- \frac{7}{4}) \cdot 22

Apply rule:

(- a) = - a

(- \frac{7}{4}) = - \frac{7}{4}

= - \frac{7}{4} \cdot 22

- \frac{7}{4} \cdot 22 = - \frac{7}{2} 2

- \frac{7}{4} \cdot 22

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

= - \frac{7}{4} \cdot \frac{2}{1} 2

Apply the fraction rule:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

\frac{7}{4} \cdot \frac{2}{1} = \frac{7 \cdot 2}{4 \cdot 1}

= - \frac{7 \cdot 2}{4 \cdot 1} 2

\frac{7 \cdot 2}{4 \cdot 1} = \frac{7}{2}

\frac{7 \cdot 2}{4 \cdot 1}

Multiply the numbers:

4 \cdot 1 = 4

= \frac{7 \cdot 2}{4}

Factor the number:

4 = 2 \cdot 2

= \frac{7 \cdot 2}{2 \cdot 2}

Cancel the common factor:

2

= \frac{7}{2}

= - \frac{7}{2} 2

= - \frac{7}{2} 2

2 a 22 - 1 = - \frac{7}{2} 2

2 a 22 - 1 = - \frac{7}{2} 2

2 a 22 - 1 = - \frac{7}{2} 2

Divide both sides by

2 22 - 1

2 a 22 - 1 = - \frac{7}{2} 2

Divide both sides by

2 22 - 1

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{- \frac{7}{2} 2}{2 2 2 - 1}

Simplify

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{- \frac{7}{2} 2}{2 2 2 - 1}

Simplify

\frac{2 a 2 2 - 1}{2 2 2 - 1} : a

\frac{2 a 2 2 - 1}{2 2 2 - 1}

Cancel the common factor:

2

= \frac{a 2 2 - 1}{2 2 - 1}

Cancel the common factor:

22 - 1

= a

Simplify

\frac{- \frac{7}{2} 2}{2 2 2 - 1} : - \frac{7}{2 2 2 2 - 1}

\frac{- \frac{7}{2} 2}{2 2 2 - 1}

Apply radical rule:

a = a a

2 = 22

= \frac{- \frac{7}{2} 2}{2 2 2 2 - 1}

Cancel the common factor:

2

= \frac{- \frac{7}{2}}{2 2 2 - 1}

\frac{- \frac{7}{2}}{2 2 2 - 1} = - \frac{7}{2 2 2 2 - 1}

\frac{- \frac{7}{2}}{2 2 2 - 1}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{7}{2}}{2 2 2 - 1}

Apply the fraction rule:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

\frac{\frac{7}{2}}{2 2 2 - 1} = \frac{7}{2 2 2 2 - 1}

= - \frac{7}{2 2 2 2 - 1}

= - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

a = - \frac{7}{2 2 2 2 - 1}

For

2 ab = - \frac{7}{4}

, subsitute

b

with

- \frac{2 2 - 1}{2 2} : a = \frac{7}{2 2 2 2 - 1}

For

2 ab = - \frac{7}{4}

, subsitute

b

with

- \frac{2 2 - 1}{2 2}

2 a (- \frac{2 2 - 1}{2 2}) = - \frac{7}{4}

Solve

2 a (- \frac{2 2 - 1}{2 2}) = - \frac{7}{4} : a = \frac{7}{2 2 2 2 - 1}

2 a (- \frac{2 2 - 1}{2 2}) = - \frac{7}{4}

Divide both sides by

2 (- \frac{2 2 - 1}{2 2})

2 a (- \frac{2 2 - 1}{2 2}) = - \frac{7}{4}

Divide both sides by

2 (- \frac{2 2 - 1}{2 2})

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} = \frac{- \frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} = \frac{- \frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} : a

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )} : \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

\frac{2 a ( - \frac{2 2 - 1}{2 2} )}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

2 a (- \frac{2 2 - 1}{2 2}) = - 2 a \frac{2 2 - 1}{2 2}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2 2}) = - 2 \cdot \frac{2 2 - 1}{2 2}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

= \frac{- 2 a \frac{2 2 - 1}{2 2}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

Cancel the common factor:

- 2

= \frac{a \frac{2 2 - 1}{2 2}}{\frac{2 2 - 1}{2 2}}

Cancel the common factor:

\frac{2 2 - 1}{2 2}

= a

Simplify

\frac{- \frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )} : \frac{7}{2 2 2 2 - 1}

\frac{- \frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{7}{4}}{2 ( - \frac{2 2 - 1}{2 2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2 2}) = - 2 \cdot \frac{2 2 - 1}{2 2}

= - \frac{\frac{7}{4}}{- 2 \cdot \frac{2 2 - 1}{2 2}}

- 2 \cdot \frac{2 2 - 1}{2 2} = - \frac{2 2 - 1}{2}

- 2 \cdot \frac{2 2 - 1}{2 2}

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

= - \frac{2}{1} \cdot \frac{2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

\frac{2}{1} \cdot \frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{1 \cdot 2 2}

= - \frac{2 2 2 - 1}{1 \cdot 2 2}

\frac{2 2 2 - 1}{1 \cdot 2 2} = \frac{2 2 - 1}{2}

\frac{2 2 2 - 1}{1 \cdot 2 2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 2 2 - 1}{2 2}

Cancel the common factor:

2

= \frac{2 2 - 1}{2}

= - \frac{2 2 - 1}{2}

= - \frac{\frac{7}{4}}{- \frac{2 2 - 1}{2}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

\frac{\frac{7}{4}}{- \frac{2 2 - 1}{2}} = - \frac{\frac{7}{4}}{\frac{2 2 - 1}{2}}

= - - \frac{\frac{7}{4}}{\frac{2 2 - 1}{2}}

Apply rule:

- (- a) = a

- - \frac{\frac{7}{4}}{\frac{2 2 - 1}{2}} = \frac{\frac{7}{4}}{\frac{2 2 - 1}{2}}

= \frac{\frac{7}{4}}{\frac{2 2 - 1}{2}}

Apply the fraction rule:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{7 2}{4 2 2 - 1}

Cancel

\frac{7 2}{4 2 2 - 1} : \frac{7}{2 \cdot 2 2 2 - 1}

\frac{7 2}{4 2 2 - 1}

Factor the number:

4 = 2 \cdot 2

= \frac{7 2}{2 \cdot 2 2 2 - 1}

Apply radical rule:

a = a a

2 = 22

= \frac{7 2}{2 2 \cdot 2 2 2 - 1}

Cancel the common factor:

2

= \frac{7}{2 \cdot 2 2 2 - 1}

= \frac{7}{2 \cdot 2 2 2 - 1}

= \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

a = \frac{7}{2 2 2 2 - 1}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = \frac{1}{4}

Remove the ones that don't agree with the equation.

Check the solution

a = \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2} :

True

a^{2} - b^{2} = \frac{1}{4}

Plug in

a = \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2}

(\frac{7}{2 2 2 2 - 1})^{2} - (- \frac{2 2 - 1}{2 2})^{2} = \frac{1}{4}

Refine

\frac{1}{4} = \frac{1}{4}

True

Check the solution

a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} :

True

a^{2} - b^{2} = \frac{1}{4}

Plug in

a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2}

(- \frac{7}{2 2 2 2 - 1})^{2} - (\frac{2 2 - 1}{2 2})^{2} = \frac{1}{4}

Refine

\frac{1}{4} = \frac{1}{4}

True

Check the solutions by plugging them into

2 ab = - \frac{7}{4}

Remove the ones that don't agree with the equation.

Check the solution

a = \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2} :

True

2 ab = - \frac{7}{4}

Plug in

a = \frac{7}{2 2 2 2 - 1}, b = - \frac{2 2 - 1}{2 2}

2 \cdot \frac{7}{2 2 2 2 - 1} (- \frac{2 2 - 1}{2 2}) = - \frac{7}{4}

Refine

- \frac{7}{4} = - \frac{7}{4}

True

Check the solution

a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} :

True

2 ab = - \frac{7}{4}

Plug in

a = - \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2}

2 (- \frac{7}{2 2 2 2 - 1}) \frac{2 2 - 1}{2 2} = - \frac{7}{4}

Refine

- \frac{7}{4} = - \frac{7}{4}

True

Therefore, the final solutions for

a^{2} - b^{2} = \frac{1}{4}, 2 ab = - \frac{7}{4}

are

a = - \frac{7}{2 2 2 2 - 1}, a = \frac{7}{2 2 2 2 - 1}, b = \frac{2 2 - 1}{2 2} b = - \frac{2 2 - 1}{2 2}

Substitute back

u = a + bi

u = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

The solutions are

u = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i, u = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, u = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

Substitute back

u = sin (x)

sin (x) = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, sin (x) = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i, sin (x) = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, sin (x) = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

sin (x) = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, sin (x) = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i, sin (x) = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i, sin (x) = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

sin (x) = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i :

No Solution

sin (x) = \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i

Simplify

\frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i : \frac{- 14 + 28 2 + 4 - 7 + 14 2}{28} + i \frac{2 - 1 + 2 2}{4}

\frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i

\frac{7}{2 2 2 2 - 1} = \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{7}{2 2 2 2 - 1}

Multiply by the conjugate

\frac{2}{2}

= \frac{7 2}{2 2 2 2 - 1 2}

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

22 22 - 1 2 = 4 22 - 1

22 22 - 1 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 22 - 1 \cdot 2^{1 + \frac{1}{2} + \frac{1}{2}}

2^{1 + \frac{1}{2} + \frac{1}{2}} = 2^{2}

2^{1 + \frac{1}{2} + \frac{1}{2}}

Combine the fractions

\frac{1}{2} + \frac{1}{2} : 1

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} 22 - 1

2^{2} = 4

= 4 22 - 1

= \frac{14}{4 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{14 2 2 - 1}{4 2 2 - 1 2 2 - 1}

4 22 - 1 22 - 1 = 82 - 4

4 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 4 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 22, c = 1

= 4 \cdot 22 - 4 \cdot 1

Simplify

4 \cdot 22 - 4 \cdot 1 : 82 - 4

4 \cdot 22 - 4 \cdot 1

Multiply the numbers:

4 \cdot 2 = 8

= 82 - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 82 - 4

= 82 - 4

= \frac{14 2 2 - 1}{8 2 - 4}

Multiply by the conjugate

\frac{8 2 + 4}{8 2 + 4}

= \frac{14 2 2 - 1 ( 8 2 + 4 )}{( 8 2 - 4 ) ( 8 2 + 4 )}

14 22 - 1 (82 + 4) = 167 22 - 1 + 414 22 - 1

14 22 - 1 (82 + 4)

= 14 (82 + 4) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 14 22 - 1, b = 82, c = 4

= 14 22 - 1 \cdot 82 + 14 22 - 1 \cdot 4

= 8142 22 - 1 + 414 22 - 1

8142 22 - 1 = 167 22 - 1

8142 22 - 1

Factor integer

8 = 2^{3}

= 2^{3} 142 22 - 1

Factor integer

14 = 2 \cdot 7

= 2^{3} 2 \cdot 7 2 22 - 1

Apply radical rule:

2 \cdot 7 = 27

= 2^{3} 272 22 - 1

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 27 22 - 1

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{3} \cdot 2 = 2^{3 + 1}

= 7 \cdot 2^{3 + 1} 22 - 1

Add the numbers:

3 + 1 = 4

= 7 \cdot 2^{4} 22 - 1

2^{4} = 16

= 167 22 - 1

= 167 22 - 1 + 414 22 - 1

(82 - 4) (82 + 4) = 112

(82 - 4) (82 + 4)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 82, b = 4

= (82)^{2} - 4^{2}

Simplify

(82)^{2} - 4^{2} : 112

(82)^{2} - 4^{2}

(82)^{2} = 128

(82)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 8^{2} \cdot 2

8^{2} = 64

= 64 \cdot 2

Multiply the numbers:

64 \cdot 2 = 128

= 128

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 128 - 16

Subtract the numbers:

128 - 16 = 112

= 112

= 112

= \frac{16 7 2 2 - 1 + 4 14 2 2 - 1}{112}

Factor

167 22 - 1 + 414 22 - 1 : 4 - 1 + 22 (47 + 14)

167 22 - 1 + 414 22 - 1

Rewrite as

= 4 \cdot 4 - 1 + 22 7 + 4 - 1 + 22 14

Factor out common term

4 - 1 + 22

= 4 - 1 + 22 (47 + 14)

= \frac{4 - 1 + 2 2 ( 4 7 + 14 )}{112}

Cancel the common factor:

4

= \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{2 2 - 1}{2 2} i = i \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= i \frac{2 2 2 - 1}{4}

= \frac{( 4 7 + 14 ) 2 2 - 1}{28} + i \frac{2 2 2 - 1}{4}

Rewrite

\frac{( 4 7 + 14 ) 2 2 - 1}{28} + \frac{2 2 2 - 1}{4} i

in standard complex form:

\frac{4 14 2 - 7 + 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} + \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{( 4 7 + 14 ) 2 2 - 1}{28}

Expand

(47 + 14) 22 - 1 : 4 142 - 7 + 282 - 14

(47 + 14) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 47, c = 14

= 22 - 1 \cdot 47 + 22 - 1 14

= 47 22 - 1 + 14 22 - 1

Simplify

47 22 - 1 + 14 22 - 1 : 4 142 - 7 + 282 - 14

47 22 - 1 + 14 22 - 1

47 22 - 1 = 4 142 - 7

47 22 - 1

Apply radical rule:

a b = a \cdot b

7 22 - 1 = 7 (22 - 1)

= 4 7 (22 - 1)

Expand

7 (22 - 1) : 142 - 7

7 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 22, c = 1

= 7 \cdot 22 - 7 \cdot 1

Simplify

7 \cdot 22 - 7 \cdot 1 : 142 - 7

7 \cdot 22 - 7 \cdot 1

Multiply the numbers:

7 \cdot 2 = 14

= 142 - 7 \cdot 1

Multiply the numbers:

7 \cdot 1 = 7

= 142 - 7

= 142 - 7

= 4 142 - 7

14 22 - 1 = 282 - 14

14 22 - 1

Apply radical rule:

a b = a \cdot b

14 22 - 1 = 14 (22 - 1)

= 14 (22 - 1)

Expand

14 (22 - 1) : 282 - 14

14 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 14, b = 22, c = 1

= 14 \cdot 22 - 14 \cdot 1

Simplify

14 \cdot 22 - 14 \cdot 1 : 282 - 14

14 \cdot 22 - 14 \cdot 1

Multiply the numbers:

14 \cdot 2 = 28

= 282 - 14 \cdot 1

Multiply the numbers:

14 \cdot 1 = 14

= 282 - 14

= 282 - 14

= 282 - 14

= 4 142 - 7 + 282 - 14

= 4 142 - 7 + 282 - 14

= \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{2 2 2 - 1}{4} i = \frac{i 2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4} i

\frac{2 2 2 - 1}{4} = \frac{2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 2 2 - 1}{2 ^{2}}

Cancel

\frac{2 2 2 - 1}{2 ^{2}} : \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

\frac{2 2 2 - 1}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 2 2 - 1}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{2 2 - 1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{2 2 - 1}{2 2}

= i \frac{2 2 - 1}{2 2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2 2}

= \frac{4 14 2 - 7 + 28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 14 2 - 7 + 28 2 - 14}{28} = \frac{4 14 2 - 7}{28} + \frac{28 2 - 14}{28}

= \frac{4 14 2 - 7}{28} + \frac{28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Cancel

\frac{4 14 2 - 7}{28} : \frac{14 2 - 7}{7}

\frac{4 14 2 - 7}{28}

Cancel the common factor:

4

= \frac{14 2 - 7}{7}

= \frac{14 2 - 7}{7} + \frac{28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Group the real part and the imaginary part of the complex number

= (\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}) + \frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= (\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}) + \frac{2 2 2 - 1}{4} i

\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}

Least Common Multiplier of

7, 28 : 28

7, 28

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

28 : 2 \cdot 2 \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

7

28

= 7 \cdot 2 \cdot 2

Multiply the numbers:

7 \cdot 2 \cdot 2 = 28

= 28

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

28

For

\frac{14 2 - 7}{7} :

multiply the denominator and numerator by

4

\frac{14 2 - 7}{7} = \frac{14 2 - 7 \cdot 4}{7 \cdot 4} = \frac{14 2 - 7 \cdot 4}{28}

= \frac{14 2 - 7 \cdot 4}{28} + \frac{28 2 - 14}{28}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{14 2 - 7 \cdot 4 + 28 2 - 14}{28}

= \frac{4 14 2 - 7 + 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

= \frac{4 14 2 - 7 + 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

No Solution

sin (x) = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i :

No Solution

sin (x) = - \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

Simplify

- \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i : \frac{- - 14 + 28 2 - 4 - 7 + 14 2}{28} - i \frac{2 - 1 + 2 2}{4}

- \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

\frac{7}{2 2 2 2 - 1} = \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{7}{2 2 2 2 - 1}

Multiply by the conjugate

\frac{2}{2}

= \frac{7 2}{2 2 2 2 - 1 2}

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

22 22 - 1 2 = 4 22 - 1

22 22 - 1 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 22 - 1 \cdot 2^{1 + \frac{1}{2} + \frac{1}{2}}

2^{1 + \frac{1}{2} + \frac{1}{2}} = 2^{2}

2^{1 + \frac{1}{2} + \frac{1}{2}}

Combine the fractions

\frac{1}{2} + \frac{1}{2} : 1

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} 22 - 1

2^{2} = 4

= 4 22 - 1

= \frac{14}{4 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{14 2 2 - 1}{4 2 2 - 1 2 2 - 1}

4 22 - 1 22 - 1 = 82 - 4

4 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 4 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 22, c = 1

= 4 \cdot 22 - 4 \cdot 1

Simplify

4 \cdot 22 - 4 \cdot 1 : 82 - 4

4 \cdot 22 - 4 \cdot 1

Multiply the numbers:

4 \cdot 2 = 8

= 82 - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 82 - 4

= 82 - 4

= \frac{14 2 2 - 1}{8 2 - 4}

Multiply by the conjugate

\frac{8 2 + 4}{8 2 + 4}

= \frac{14 2 2 - 1 ( 8 2 + 4 )}{( 8 2 - 4 ) ( 8 2 + 4 )}

14 22 - 1 (82 + 4) = 167 22 - 1 + 414 22 - 1

14 22 - 1 (82 + 4)

= 14 (82 + 4) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 14 22 - 1, b = 82, c = 4

= 14 22 - 1 \cdot 82 + 14 22 - 1 \cdot 4

= 8142 22 - 1 + 414 22 - 1

8142 22 - 1 = 167 22 - 1

8142 22 - 1

Factor integer

8 = 2^{3}

= 2^{3} 142 22 - 1

Factor integer

14 = 2 \cdot 7

= 2^{3} 2 \cdot 7 2 22 - 1

Apply radical rule:

2 \cdot 7 = 27

= 2^{3} 272 22 - 1

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 27 22 - 1

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{3} \cdot 2 = 2^{3 + 1}

= 7 \cdot 2^{3 + 1} 22 - 1

Add the numbers:

3 + 1 = 4

= 7 \cdot 2^{4} 22 - 1

2^{4} = 16

= 167 22 - 1

= 167 22 - 1 + 414 22 - 1

(82 - 4) (82 + 4) = 112

(82 - 4) (82 + 4)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 82, b = 4

= (82)^{2} - 4^{2}

Simplify

(82)^{2} - 4^{2} : 112

(82)^{2} - 4^{2}

(82)^{2} = 128

(82)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 8^{2} \cdot 2

8^{2} = 64

= 64 \cdot 2

Multiply the numbers:

64 \cdot 2 = 128

= 128

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 128 - 16

Subtract the numbers:

128 - 16 = 112

= 112

= 112

= \frac{16 7 2 2 - 1 + 4 14 2 2 - 1}{112}

Factor

167 22 - 1 + 414 22 - 1 : 4 - 1 + 22 (47 + 14)

167 22 - 1 + 414 22 - 1

Rewrite as

= 4 \cdot 4 - 1 + 22 7 + 4 - 1 + 22 14

Factor out common term

4 - 1 + 22

= 4 - 1 + 22 (47 + 14)

= \frac{4 - 1 + 2 2 ( 4 7 + 14 )}{112}

Cancel the common factor:

4

= \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{2 2 - 1}{2 2} i = i \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= i \frac{2 2 2 - 1}{4}

= - \frac{( 4 7 + 14 ) 2 2 - 1}{28} - i \frac{2 2 2 - 1}{4}

Rewrite

- \frac{( 4 7 + 14 ) 2 2 - 1}{28} - \frac{2 2 2 - 1}{4} i

in standard complex form:

\frac{- 4 14 2 - 7 - 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

- \frac{( 4 7 + 14 ) 2 2 - 1}{28} - \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{( 4 7 + 14 ) 2 2 - 1}{28}

Expand

(47 + 14) 22 - 1 : 4 142 - 7 + 282 - 14

(47 + 14) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 47, c = 14

= 22 - 1 \cdot 47 + 22 - 1 14

= 47 22 - 1 + 14 22 - 1

Simplify

47 22 - 1 + 14 22 - 1 : 4 142 - 7 + 282 - 14

47 22 - 1 + 14 22 - 1

47 22 - 1 = 4 142 - 7

47 22 - 1

Apply radical rule:

a b = a \cdot b

7 22 - 1 = 7 (22 - 1)

= 4 7 (22 - 1)

Expand

7 (22 - 1) : 142 - 7

7 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 22, c = 1

= 7 \cdot 22 - 7 \cdot 1

Simplify

7 \cdot 22 - 7 \cdot 1 : 142 - 7

7 \cdot 22 - 7 \cdot 1

Multiply the numbers:

7 \cdot 2 = 14

= 142 - 7 \cdot 1

Multiply the numbers:

7 \cdot 1 = 7

= 142 - 7

= 142 - 7

= 4 142 - 7

14 22 - 1 = 282 - 14

14 22 - 1

Apply radical rule:

a b = a \cdot b

14 22 - 1 = 14 (22 - 1)

= 14 (22 - 1)

Expand

14 (22 - 1) : 282 - 14

14 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 14, b = 22, c = 1

= 14 \cdot 22 - 14 \cdot 1

Simplify

14 \cdot 22 - 14 \cdot 1 : 282 - 14

14 \cdot 22 - 14 \cdot 1

Multiply the numbers:

14 \cdot 2 = 28

= 282 - 14 \cdot 1

Multiply the numbers:

14 \cdot 1 = 14

= 282 - 14

= 282 - 14

= 282 - 14

= 4 142 - 7 + 282 - 14

= 4 142 - 7 + 282 - 14

= \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{2 2 2 - 1}{4} i = \frac{i 2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4} i

\frac{2 2 2 - 1}{4} = \frac{2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 2 2 - 1}{2 ^{2}}

Cancel

\frac{2 2 2 - 1}{2 ^{2}} : \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

\frac{2 2 2 - 1}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 2 2 - 1}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{2 2 - 1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{2 2 - 1}{2 2}

= i \frac{2 2 - 1}{2 2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2 2}

= - \frac{4 14 2 - 7 + 28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 14 2 - 7 + 28 2 - 14}{28} = - (\frac{4 14 2 - 7}{28}) - (\frac{28 2 - 14}{28})

= - (\frac{4 14 2 - 7}{28}) - (\frac{28 2 - 14}{28}) - \frac{i 2 2 - 1}{2 2}

Remove parentheses:

(a) = a

= - \frac{4 14 2 - 7}{28} - \frac{28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Cancel

\frac{4 14 2 - 7}{28} : \frac{14 2 - 7}{7}

\frac{4 14 2 - 7}{28}

Cancel the common factor:

4

= \frac{14 2 - 7}{7}

= - \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Group the real part and the imaginary part of the complex number

= (- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}) - \frac{2 2 - 1}{2 2} i

- \frac{2 2 - 1}{2 2} = - \frac{2 2 2 - 1}{4}

- \frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 2 2 - 1}{4}

= (- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}) - \frac{2 2 2 - 1}{4} i

- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28} = \frac{- 4 14 2 - 7 - 28 2 - 14}{28}

- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}

Least Common Multiplier of

7, 28 : 28

7, 28

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

28 : 2 \cdot 2 \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

7

28

= 7 \cdot 2 \cdot 2

Multiply the numbers:

7 \cdot 2 \cdot 2 = 28

= 28

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

28

For

\frac{14 2 - 7}{7} :

multiply the denominator and numerator by

4

\frac{14 2 - 7}{7} = \frac{14 2 - 7 \cdot 4}{7 \cdot 4} = \frac{14 2 - 7 \cdot 4}{28}

= - \frac{14 2 - 7 \cdot 4}{28} - \frac{28 2 - 14}{28}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 14 2 - 7 \cdot 4 - 28 2 - 14}{28}

= \frac{- 4 14 2 - 7 - 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

= \frac{- 4 14 2 - 7 - 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

No Solution

sin (x) = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i :

No Solution

sin (x) = - \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i

Simplify

- \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i : \frac{- - 14 + 28 2 - 4 - 7 + 14 2}{28} + i \frac{2 - 1 + 2 2}{4}

- \frac{7}{2 2 2 2 - 1} + \frac{2 2 - 1}{2 2} i

\frac{7}{2 2 2 2 - 1} = \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{7}{2 2 2 2 - 1}

Multiply by the conjugate

\frac{2}{2}

= \frac{7 2}{2 2 2 2 - 1 2}

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

22 22 - 1 2 = 4 22 - 1

22 22 - 1 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 22 - 1 \cdot 2^{1 + \frac{1}{2} + \frac{1}{2}}

2^{1 + \frac{1}{2} + \frac{1}{2}} = 2^{2}

2^{1 + \frac{1}{2} + \frac{1}{2}}

Combine the fractions

\frac{1}{2} + \frac{1}{2} : 1

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} 22 - 1

2^{2} = 4

= 4 22 - 1

= \frac{14}{4 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{14 2 2 - 1}{4 2 2 - 1 2 2 - 1}

4 22 - 1 22 - 1 = 82 - 4

4 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 4 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 22, c = 1

= 4 \cdot 22 - 4 \cdot 1

Simplify

4 \cdot 22 - 4 \cdot 1 : 82 - 4

4 \cdot 22 - 4 \cdot 1

Multiply the numbers:

4 \cdot 2 = 8

= 82 - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 82 - 4

= 82 - 4

= \frac{14 2 2 - 1}{8 2 - 4}

Multiply by the conjugate

\frac{8 2 + 4}{8 2 + 4}

= \frac{14 2 2 - 1 ( 8 2 + 4 )}{( 8 2 - 4 ) ( 8 2 + 4 )}

14 22 - 1 (82 + 4) = 167 22 - 1 + 414 22 - 1

14 22 - 1 (82 + 4)

= 14 (82 + 4) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 14 22 - 1, b = 82, c = 4

= 14 22 - 1 \cdot 82 + 14 22 - 1 \cdot 4

= 8142 22 - 1 + 414 22 - 1

8142 22 - 1 = 167 22 - 1

8142 22 - 1

Factor integer

8 = 2^{3}

= 2^{3} 142 22 - 1

Factor integer

14 = 2 \cdot 7

= 2^{3} 2 \cdot 7 2 22 - 1

Apply radical rule:

2 \cdot 7 = 27

= 2^{3} 272 22 - 1

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 27 22 - 1

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{3} \cdot 2 = 2^{3 + 1}

= 7 \cdot 2^{3 + 1} 22 - 1

Add the numbers:

3 + 1 = 4

= 7 \cdot 2^{4} 22 - 1

2^{4} = 16

= 167 22 - 1

= 167 22 - 1 + 414 22 - 1

(82 - 4) (82 + 4) = 112

(82 - 4) (82 + 4)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 82, b = 4

= (82)^{2} - 4^{2}

Simplify

(82)^{2} - 4^{2} : 112

(82)^{2} - 4^{2}

(82)^{2} = 128

(82)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 8^{2} \cdot 2

8^{2} = 64

= 64 \cdot 2

Multiply the numbers:

64 \cdot 2 = 128

= 128

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 128 - 16

Subtract the numbers:

128 - 16 = 112

= 112

= 112

= \frac{16 7 2 2 - 1 + 4 14 2 2 - 1}{112}

Factor

167 22 - 1 + 414 22 - 1 : 4 - 1 + 22 (47 + 14)

167 22 - 1 + 414 22 - 1

Rewrite as

= 4 \cdot 4 - 1 + 22 7 + 4 - 1 + 22 14

Factor out common term

4 - 1 + 22

= 4 - 1 + 22 (47 + 14)

= \frac{4 - 1 + 2 2 ( 4 7 + 14 )}{112}

Cancel the common factor:

4

= \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{2 2 - 1}{2 2} i = i \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= i \frac{2 2 2 - 1}{4}

= - \frac{( 4 7 + 14 ) 2 2 - 1}{28} + i \frac{2 2 2 - 1}{4}

Rewrite

- \frac{( 4 7 + 14 ) 2 2 - 1}{28} + \frac{2 2 2 - 1}{4} i

in standard complex form:

\frac{- 4 14 2 - 7 - 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

- \frac{( 4 7 + 14 ) 2 2 - 1}{28} + \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{( 4 7 + 14 ) 2 2 - 1}{28}

Expand

(47 + 14) 22 - 1 : 4 142 - 7 + 282 - 14

(47 + 14) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 47, c = 14

= 22 - 1 \cdot 47 + 22 - 1 14

= 47 22 - 1 + 14 22 - 1

Simplify

47 22 - 1 + 14 22 - 1 : 4 142 - 7 + 282 - 14

47 22 - 1 + 14 22 - 1

47 22 - 1 = 4 142 - 7

47 22 - 1

Apply radical rule:

a b = a \cdot b

7 22 - 1 = 7 (22 - 1)

= 4 7 (22 - 1)

Expand

7 (22 - 1) : 142 - 7

7 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 22, c = 1

= 7 \cdot 22 - 7 \cdot 1

Simplify

7 \cdot 22 - 7 \cdot 1 : 142 - 7

7 \cdot 22 - 7 \cdot 1

Multiply the numbers:

7 \cdot 2 = 14

= 142 - 7 \cdot 1

Multiply the numbers:

7 \cdot 1 = 7

= 142 - 7

= 142 - 7

= 4 142 - 7

14 22 - 1 = 282 - 14

14 22 - 1

Apply radical rule:

a b = a \cdot b

14 22 - 1 = 14 (22 - 1)

= 14 (22 - 1)

Expand

14 (22 - 1) : 282 - 14

14 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 14, b = 22, c = 1

= 14 \cdot 22 - 14 \cdot 1

Simplify

14 \cdot 22 - 14 \cdot 1 : 282 - 14

14 \cdot 22 - 14 \cdot 1

Multiply the numbers:

14 \cdot 2 = 28

= 282 - 14 \cdot 1

Multiply the numbers:

14 \cdot 1 = 14

= 282 - 14

= 282 - 14

= 282 - 14

= 4 142 - 7 + 282 - 14

= 4 142 - 7 + 282 - 14

= \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{2 2 2 - 1}{4} i = \frac{i 2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4} i

\frac{2 2 2 - 1}{4} = \frac{2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 2 2 - 1}{2 ^{2}}

Cancel

\frac{2 2 2 - 1}{2 ^{2}} : \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

\frac{2 2 2 - 1}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 2 2 - 1}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{2 2 - 1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{2 2 - 1}{2 2}

= i \frac{2 2 - 1}{2 2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2 2}

= - \frac{4 14 2 - 7 + 28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 14 2 - 7 + 28 2 - 14}{28} = - (\frac{4 14 2 - 7}{28}) - (\frac{28 2 - 14}{28})

= - (\frac{4 14 2 - 7}{28}) - (\frac{28 2 - 14}{28}) + \frac{i 2 2 - 1}{2 2}

Remove parentheses:

(a) = a

= - \frac{4 14 2 - 7}{28} - \frac{28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Cancel

\frac{4 14 2 - 7}{28} : \frac{14 2 - 7}{7}

\frac{4 14 2 - 7}{28}

Cancel the common factor:

4

= \frac{14 2 - 7}{7}

= - \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28} + \frac{i 2 2 - 1}{2 2}

Group the real part and the imaginary part of the complex number

= (- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}) + \frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= (- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}) + \frac{2 2 2 - 1}{4} i

- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28} = \frac{- 4 14 2 - 7 - 28 2 - 14}{28}

- \frac{14 2 - 7}{7} - \frac{28 2 - 14}{28}

Least Common Multiplier of

7, 28 : 28

7, 28

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

28 : 2 \cdot 2 \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

7

28

= 7 \cdot 2 \cdot 2

Multiply the numbers:

7 \cdot 2 \cdot 2 = 28

= 28

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

28

For

\frac{14 2 - 7}{7} :

multiply the denominator and numerator by

4

\frac{14 2 - 7}{7} = \frac{14 2 - 7 \cdot 4}{7 \cdot 4} = \frac{14 2 - 7 \cdot 4}{28}

= - \frac{14 2 - 7 \cdot 4}{28} - \frac{28 2 - 14}{28}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 14 2 - 7 \cdot 4 - 28 2 - 14}{28}

= \frac{- 4 14 2 - 7 - 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

= \frac{- 4 14 2 - 7 - 28 2 - 14}{28} + \frac{2 2 2 - 1}{4} i

No Solution

sin (x) = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i :

No Solution

sin (x) = \frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

Simplify

\frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i : \frac{- 14 + 28 2 + 4 - 7 + 14 2}{28} - i \frac{2 - 1 + 2 2}{4}

\frac{7}{2 2 2 2 - 1} - \frac{2 2 - 1}{2 2} i

\frac{7}{2 2 2 2 - 1} = \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{7}{2 2 2 2 - 1}

Multiply by the conjugate

\frac{2}{2}

= \frac{7 2}{2 2 2 2 - 1 2}

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

22 22 - 1 2 = 4 22 - 1

22 22 - 1 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 22 - 1 \cdot 2^{1 + \frac{1}{2} + \frac{1}{2}}

2^{1 + \frac{1}{2} + \frac{1}{2}} = 2^{2}

2^{1 + \frac{1}{2} + \frac{1}{2}}

Combine the fractions

\frac{1}{2} + \frac{1}{2} : 1

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} 22 - 1

2^{2} = 4

= 4 22 - 1

= \frac{14}{4 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{14 2 2 - 1}{4 2 2 - 1 2 2 - 1}

4 22 - 1 22 - 1 = 82 - 4

4 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 4 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 22, c = 1

= 4 \cdot 22 - 4 \cdot 1

Simplify

4 \cdot 22 - 4 \cdot 1 : 82 - 4

4 \cdot 22 - 4 \cdot 1

Multiply the numbers:

4 \cdot 2 = 8

= 82 - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 82 - 4

= 82 - 4

= \frac{14 2 2 - 1}{8 2 - 4}

Multiply by the conjugate

\frac{8 2 + 4}{8 2 + 4}

= \frac{14 2 2 - 1 ( 8 2 + 4 )}{( 8 2 - 4 ) ( 8 2 + 4 )}

14 22 - 1 (82 + 4) = 167 22 - 1 + 414 22 - 1

14 22 - 1 (82 + 4)

= 14 (82 + 4) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 14 22 - 1, b = 82, c = 4

= 14 22 - 1 \cdot 82 + 14 22 - 1 \cdot 4

= 8142 22 - 1 + 414 22 - 1

8142 22 - 1 = 167 22 - 1

8142 22 - 1

Factor integer

8 = 2^{3}

= 2^{3} 142 22 - 1

Factor integer

14 = 2 \cdot 7

= 2^{3} 2 \cdot 7 2 22 - 1

Apply radical rule:

2 \cdot 7 = 27

= 2^{3} 272 22 - 1

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 27 22 - 1

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{3} \cdot 2 = 2^{3 + 1}

= 7 \cdot 2^{3 + 1} 22 - 1

Add the numbers:

3 + 1 = 4

= 7 \cdot 2^{4} 22 - 1

2^{4} = 16

= 167 22 - 1

= 167 22 - 1 + 414 22 - 1

(82 - 4) (82 + 4) = 112

(82 - 4) (82 + 4)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 82, b = 4

= (82)^{2} - 4^{2}

Simplify

(82)^{2} - 4^{2} : 112

(82)^{2} - 4^{2}

(82)^{2} = 128

(82)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 8^{2} \cdot 2

8^{2} = 64

= 64 \cdot 2

Multiply the numbers:

64 \cdot 2 = 128

= 128

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 128 - 16

Subtract the numbers:

128 - 16 = 112

= 112

= 112

= \frac{16 7 2 2 - 1 + 4 14 2 2 - 1}{112}

Factor

167 22 - 1 + 414 22 - 1 : 4 - 1 + 22 (47 + 14)

167 22 - 1 + 414 22 - 1

Rewrite as

= 4 \cdot 4 - 1 + 22 7 + 4 - 1 + 22 14

Factor out common term

4 - 1 + 22

= 4 - 1 + 22 (47 + 14)

= \frac{4 - 1 + 2 2 ( 4 7 + 14 )}{112}

Cancel the common factor:

4

= \frac{( 4 7 + 14 ) 2 2 - 1}{28}

\frac{2 2 - 1}{2 2} i = i \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2} i

\frac{2 2 - 1}{2 2} = \frac{2 2 2 - 1}{4}

\frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 2 2 - 1}{4}

= i \frac{2 2 2 - 1}{4}

= \frac{( 4 7 + 14 ) 2 2 - 1}{28} - i \frac{2 2 2 - 1}{4}

Rewrite

\frac{( 4 7 + 14 ) 2 2 - 1}{28} - \frac{2 2 2 - 1}{4} i

in standard complex form:

\frac{4 14 2 - 7 + 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} - \frac{2 2 2 - 1}{4} i

\frac{( 4 7 + 14 ) 2 2 - 1}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{( 4 7 + 14 ) 2 2 - 1}{28}

Expand

(47 + 14) 22 - 1 : 4 142 - 7 + 282 - 14

(47 + 14) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 47, c = 14

= 22 - 1 \cdot 47 + 22 - 1 14

= 47 22 - 1 + 14 22 - 1

Simplify

47 22 - 1 + 14 22 - 1 : 4 142 - 7 + 282 - 14

47 22 - 1 + 14 22 - 1

47 22 - 1 = 4 142 - 7

47 22 - 1

Apply radical rule:

a b = a \cdot b

7 22 - 1 = 7 (22 - 1)

= 4 7 (22 - 1)

Expand

7 (22 - 1) : 142 - 7

7 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 22, c = 1

= 7 \cdot 22 - 7 \cdot 1

Simplify

7 \cdot 22 - 7 \cdot 1 : 142 - 7

7 \cdot 22 - 7 \cdot 1

Multiply the numbers:

7 \cdot 2 = 14

= 142 - 7 \cdot 1

Multiply the numbers:

7 \cdot 1 = 7

= 142 - 7

= 142 - 7

= 4 142 - 7

14 22 - 1 = 282 - 14

14 22 - 1

Apply radical rule:

a b = a \cdot b

14 22 - 1 = 14 (22 - 1)

= 14 (22 - 1)

Expand

14 (22 - 1) : 282 - 14

14 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 14, b = 22, c = 1

= 14 \cdot 22 - 14 \cdot 1

Simplify

14 \cdot 22 - 14 \cdot 1 : 282 - 14

14 \cdot 22 - 14 \cdot 1

Multiply the numbers:

14 \cdot 2 = 28

= 282 - 14 \cdot 1

Multiply the numbers:

14 \cdot 1 = 14

= 282 - 14

= 282 - 14

= 282 - 14

= 4 142 - 7 + 282 - 14

= 4 142 - 7 + 282 - 14

= \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{2 2 2 - 1}{4} i = \frac{i 2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4} i

\frac{2 2 2 - 1}{4} = \frac{2 2 - 1}{2 2}

\frac{2 2 2 - 1}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 2 2 - 1}{2 ^{2}}

Cancel

\frac{2 2 2 - 1}{2 ^{2}} : \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

\frac{2 2 2 - 1}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 2 2 - 1}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{2 2 - 1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

= \frac{2 2 - 1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{2 2 - 1}{2 2}

= i \frac{2 2 - 1}{2 2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2 2}

= \frac{4 14 2 - 7 + 28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 14 2 - 7 + 28 2 - 14}{28} = \frac{4 14 2 - 7}{28} + \frac{28 2 - 14}{28}

= \frac{4 14 2 - 7}{28} + \frac{28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Cancel

\frac{4 14 2 - 7}{28} : \frac{14 2 - 7}{7}

\frac{4 14 2 - 7}{28}

Cancel the common factor:

4

= \frac{14 2 - 7}{7}

= \frac{14 2 - 7}{7} + \frac{28 2 - 14}{28} - \frac{i 2 2 - 1}{2 2}

Group the real part and the imaginary part of the complex number

= (\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}) - \frac{2 2 - 1}{2 2} i

- \frac{2 2 - 1}{2 2} = - \frac{2 2 2 - 1}{4}

- \frac{2 2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{2 2 - 1 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 2 2 - 1}{4}

= (\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}) - \frac{2 2 2 - 1}{4} i

\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28} = \frac{4 14 2 - 7 + 28 2 - 14}{28}

\frac{14 2 - 7}{7} + \frac{28 2 - 14}{28}

Least Common Multiplier of

7, 28 : 28

7, 28

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

28 : 2 \cdot 2 \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

7

28

= 7 \cdot 2 \cdot 2

Multiply the numbers:

7 \cdot 2 \cdot 2 = 28

= 28

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

28

For

\frac{14 2 - 7}{7} :

multiply the denominator and numerator by

4

\frac{14 2 - 7}{7} = \frac{14 2 - 7 \cdot 4}{7 \cdot 4} = \frac{14 2 - 7 \cdot 4}{28}

= \frac{14 2 - 7 \cdot 4}{28} + \frac{28 2 - 14}{28}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{14 2 - 7 \cdot 4 + 28 2 - 14}{28}

= \frac{4 14 2 - 7 + 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

= \frac{4 14 2 - 7 + 28 2 - 14}{28} - \frac{2 2 2 - 1}{4} i

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sec^4(x)=sec^2(x)tan^2(x)-2tan^4(x) ?
The general solution for sec^4(x)=sec^2(x)tan^2(x)-2tan^4(x) is No Solution for x\in\mathbb{R}