What is the general solution for sin^4(x)=-1/8 ?

The general solution for sin^4(x)=-1/8 is No Solution for x\in\mathbb{R}

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sin^4(x)=-1/8

Solution

sin^{4} (x) = - \frac{1}{8}

Solution

No Solution for x \in R

Solution steps

sin^{4} (x) = - \frac{1}{8}

Solve by substitution

sin^{4} (x) = - \frac{1}{8}

Let:

sin (x) = w

w^{4} = - \frac{1}{8}

w^{4} = - \frac{1}{8}

Rewrite the equation with

u = w^{2}

and

u^{2} = w^{4}

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

Solve

u^{2} = - \frac{1}{8} : u = i \frac{2}{4}, u = - i \frac{2}{4}

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

For

(g (x))^{2} = f (a)

the solutions are

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

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

Simplify

- \frac{1}{8} : i \frac{2}{4}

- \frac{1}{8}

Apply radical rule:

- a = - 1 a

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

= - 1 \frac{1}{8}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{8} = \frac{1}{8}

= i \frac{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^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{1}{2 2}

Apply rule

1 = 1

= i \frac{1}{2 2}

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

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 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}{4}

= i \frac{2}{4}

Rewrite

i \frac{2}{4}

in standard complex form:

\frac{2}{4} i

i \frac{2}{4}

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

\frac{2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

= \frac{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{1}{2 2}

= i \frac{1}{2 2}

Multiply fractions:

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

= \frac{1 i}{2 2}

Multiply:

1 i = i

= \frac{i}{2 2}

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

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 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}{4}

= \frac{2}{4} i

= \frac{2}{4} i

Simplify

- - \frac{1}{8} : - i \frac{2}{4}

- - \frac{1}{8}

Simplify

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

- \frac{1}{8}

Apply radical rule:

- a = - 1 a

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

= - 1 \frac{1}{8}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{8} = \frac{1}{8}

= i \frac{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^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{1}{2 2}

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

Apply rule

1 = 1

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

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

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 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}{4}

= - \frac{2}{4} i

u = i \frac{2}{4}, u = - i \frac{2}{4}

u = i \frac{2}{4}, u = - i \frac{2}{4}

Substitute back

u = w^{2},

solve for

w

Solve

w^{2} = i \frac{2}{4}

Substitute

w = u + v i

(u + v i)^{2} = i \frac{2}{4}

Expand

(u + v i)^{2} : (u^{2} - v^{2}) + 2 i uv

(u + v i)^{2}

= (u + i v)^{2}

Apply Perfect Square Formula:

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

a = u, b = v i

= u^{2} + 2 uv i + (v i)^{2}

(v i)^{2} = - v^{2}

(v i)^{2}

Apply exponent rule:

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

= i^{2} v^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) v^{2}

Refine

= - v^{2}

= u^{2} + 2 i uv - v^{2}

Rewrite

u^{2} + 2 i uv - v^{2}

in standard complex form:

(u^{2} - v^{2}) + 2 uv i

u^{2} + 2 i uv - v^{2}

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

= (u^{2} - v^{2}) + 2 uv i

= (u^{2} - v^{2}) + 2 uv i

(u^{2} - v^{2}) + 2 i uv = i \frac{2}{4}

Rewrite

i \frac{2}{4}

in standard complex form:

0 + \frac{2}{4} i

(u^{2} - v^{2}) + 2 i uv = 0 + \frac{2}{4} i

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

[u^{2} - v^{2} = 0 2 uv = \frac{2}{4}]

[u^{2} - v^{2} = 0 2 uv = \frac{2}{4}]

Isolate

u

for

2 uv = \frac{2}{4} : u = \frac{1}{2 ^{\frac{5}{2}} v}

2 uv = \frac{2}{4}

Factor the number:

4 = 2 \cdot 2

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

Apply radical rule:

a = a a

2 = 22

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

Cancel the common factor:

2

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

2 uv = \frac{1}{2 2}

Divide both sides by

2 v

2 uv = \frac{1}{2 2}

Divide both sides by

2 v

\frac{2 uv}{2 v} = \frac{\frac{1}{2 2}}{2 v}

Simplify

\frac{2 uv}{2 v} = \frac{\frac{1}{2 2}}{2 v}

Simplify

\frac{2 uv}{2 v} : u

\frac{2 uv}{2 v}

Cancel the common factor:

2

= \frac{uv}{v}

Cancel the common factor:

v

= u

Simplify

\frac{\frac{1}{2 2}}{2 v} : \frac{1}{2 ^{\frac{5}{2}} v}

\frac{\frac{1}{2 2}}{2 v}

Apply the fraction rule:

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

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

Simplify

22 \cdot 2 v : 2^{\frac{5}{2}} v

22 \cdot 2 v

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} 2 v

Apply radical rule:

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

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

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

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}} v

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

u = \frac{1}{2 ^{\frac{5}{2}} v}

u = \frac{1}{2 ^{\frac{5}{2}} v}

u = \frac{1}{2 ^{\frac{5}{2}} v}

Plug the solutions

u = \frac{1}{2 ^{\frac{5}{2}} v}

into

u^{2} - v^{2} = 0

For

u^{2} - v^{2} = 0

, subsitute

u

with

For

u^{2} - v^{2} = 0

, subsitute

u

with

\frac{1}{2 ^{\frac{5}{2}} v}

(\frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2} = 0

Solve

(\frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2} = 0

Simplify

(\frac{1}{2 ^{\frac{5}{2}} v})^{2} : \frac{1}{32 v ^{2}}

(\frac{1}{2 ^{\frac{5}{2}} v})^{2}

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

\frac{1}{2 ^{\frac{5}{2}} v}

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

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 2^{2} 2

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

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

Apply exponent rule:

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

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

Apply exponent rule:

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

(2^{2} 2 v)^{2} = (2^{2})^{2} (2)^{2} v^{2}

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

(2^{2})^{2} : 2^{4}

Apply exponent rule:

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

= 2^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 2^{4}

= \frac{1 ^{2}}{2 ^{4} ( 2 ) ^{2} v ^{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

= \frac{1 ^{2}}{2 ^{4} \cdot 2 v ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{4} \cdot 2 v ^{2}}

2^{4} \cdot 2 v^{2} = 2^{5} v^{2}

2^{4} \cdot 2 v^{2}

Apply exponent rule:

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

2^{4} \cdot 2 = 2^{4 + 1}

= 2^{4 + 1} v^{2}

Add the numbers:

4 + 1 = 5

= 2^{5} v^{2}

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

2^{5} = 32

= \frac{1}{32 v ^{2}}

\frac{1}{32 v ^{2}} - v^{2} = 0

Multiply both sides by

32 v^{2}

\frac{1}{32 v ^{2}} - v^{2} = 0

Multiply both sides by

32 v^{2}

\frac{1}{32 v ^{2}} \cdot 32 v^{2} - v^{2} \cdot 32 v^{2} = 0 \cdot 32 v^{2}

Simplify

\frac{1}{32 v ^{2}} \cdot 32 v^{2} - v^{2} \cdot 32 v^{2} = 0 \cdot 32 v^{2}

Simplify

\frac{1}{32 v ^{2}} \cdot 32 v^{2} : 1

\frac{1}{32 v ^{2}} \cdot 32 v^{2}

Multiply fractions:

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

= \frac{1 \cdot 32 v ^{2}}{32 v ^{2}}

Cancel the common factor:

32

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

Cancel the common factor:

v^{2}

= 1

Simplify

- v^{2} \cdot 32 v^{2} : - 32 v^{4}

- v^{2} \cdot 32 v^{2}

Apply exponent rule:

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

v^{2} v^{2} = v^{2 + 2}

= - 32 v^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 32 v^{4}

Simplify

0 \cdot 32 v^{2} : 0

0 \cdot 32 v^{2}

Apply rule

0 \cdot a = 0

= 0

1 - 32 v^{4} = 0

1 - 32 v^{4} = 0

1 - 32 v^{4} = 0

Solve

1 - 32 v^{4} = 0

Move

1

to the right side

1 - 32 v^{4} = 0

Subtract

1

from both sides

1 - 32 v^{4} - 1 = 0 - 1

Simplify

- 32 v^{4} = - 1

- 32 v^{4} = - 1

Divide both sides by

- 32

- 32 v^{4} = - 1

Divide both sides by

- 32

\frac{- 32 v ^{4}}{- 32} = \frac{- 1}{- 32}

Simplify

v^{4} = \frac{1}{32}

v^{4} = \frac{1}{32}

For

x^{n} = f (a)

, n is even, the solutions are

Apply radical rule:

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 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

= 2^{5}

Apply exponent rule:

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

2^{5} = 2^{4} \cdot 2

Apply radical rule:

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 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

= 2^{5}

Apply exponent rule:

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

2^{5} = 2^{4} \cdot 2

Apply radical rule:

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

(\frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2}

and compare to zero

Solve

2^{\frac{5}{2}} v = 0 : v = 0

2^{\frac{5}{2}} v = 0

Divide both sides by

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

2^{\frac{5}{2}} v = 0

Divide both sides by

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

\frac{2 ^{\frac{5}{2}} v}{2 ^{\frac{5}{2}}} = \frac{0}{2 ^{\frac{5}{2}}}

Simplify

\frac{2 ^{\frac{5}{2}} v}{2 ^{\frac{5}{2}}} = \frac{0}{2 ^{\frac{5}{2}}}

Simplify

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

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

Cancel the common factor:

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

= v

Simplify

\frac{0}{2 ^{\frac{5}{2}}} : 0

\frac{0}{2 ^{\frac{5}{2}}}

Apply rule

\frac{0}{a} = 0

a \neq = 0

= 0

v = 0

v = 0

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

Plug the solutions

into

2 uv = \frac{2}{4}

For

2 uv = \frac{2}{4}

, subsitute

v

with

For

2 uv = \frac{2}{4}

, subsitute

v

with

Solve

Factor the number:

4 = 2 \cdot 2

Apply radical rule:

a = a a

2 = 22

Cancel the common factor:

2

Multiply both sides by

Simplify

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}

Apply radical rule:

2^{2} \cdot 2^{\frac{1}{4}} = 2^{\frac{9}{4}}

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

Apply exponent rule:

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

2^{2} \cdot 2^{\frac{1}{4}} = 2^{2 + \frac{1}{4}}

= 2^{2 + \frac{1}{4}}

2 + \frac{1}{4} = \frac{9}{4}

2 + \frac{1}{4}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 4 + 1 = 9

2 \cdot 4 + 1

Multiply the numbers:

2 \cdot 4 = 8

= 8 + 1

Add the numbers:

8 + 1 = 9

= 9

= \frac{9}{4}

= 2^{\frac{9}{4}}

Apply the fraction rule:

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

Apply rule:

a \cdot 1 = a

2^{\frac{9}{4}} u \cdot 1 = 2^{\frac{9}{4}} u

Cancel

Simplify

\frac{2 ^{\frac{9}{4}}}{2} : 2^{\frac{5}{4}}

\frac{2 ^{\frac{9}{4}}}{2}

Apply exponent rule:

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

= 2^{\frac{9}{4} - 1}

\frac{9}{4} - 1 = \frac{5}{4}

\frac{9}{4} - 1

Convert element to fraction:

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

= - \frac{1 \cdot 4}{4} + \frac{9}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 4 + 9}{4}

- 1 \cdot 4 + 9 = 5

- 1 \cdot 4 + 9

Multiply the numbers:

1 \cdot 4 = 4

= - 4 + 9

Add/Subtract the numbers:

- 4 + 9 = 5

= 5

= \frac{5}{4}

= 2^{\frac{5}{4}}

Apply radical rule:

= \frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}}

Simplify

\frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}} : 2

\frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}}

Apply exponent rule:

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

= 2^{\frac{5}{4} - \frac{1}{4}}

\frac{5}{4} - \frac{1}{4} = 1

\frac{5}{4} - \frac{1}{4}

Apply rule

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

= \frac{5 - 1}{4}

Subtract the numbers:

5 - 1 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply exponent rule:

a^{1} = a

= 2

= 2

= 2 u

= 2 u

Simplify

Multiply the numbers:

1 \cdot 2 = 2

Cancel the common factor:

2

Apply radical rule:

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

Apply radical rule:

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

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

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

Simplify

\frac{2 ^{\frac{1}{4}}}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

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

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 - 1}{4}

Subtract the numbers:

2 - 1 = 1

= \frac{1}{4}

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

Apply exponent rule:

Divide both sides by

2

Divide both sides by

2

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

Apply the fraction rule:

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

For

2 uv = \frac{2}{4}

, subsitute

v

with

For

2 uv = \frac{2}{4}

, subsitute

v

with

Solve

Factor the number:

4 = 2 \cdot 2

Apply radical rule:

a = a a

2 = 22

Cancel the common factor:

2

Divide both sides by

Simplify

Apply rule:

a (- b) = - ab

Apply rule:

a (- b) = - ab

Cancel the common factor:

- 2

Cancel the common factor:

= u

Simplify

Apply the fraction rule:

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

Apply rule:

a (- b) = - ab

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}

Apply radical rule:

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

2 = 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}}

Apply the fraction rule:

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

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

u^{2} - v^{2} = 0

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

Check the solution

True

u^{2} - v^{2} = 0

Plug in

Refine

0 = 0

True

Check the solution

True

u^{2} - v^{2} = 0

Plug in

Refine

0 = 0

True

Check the solutions by plugging them into

2 uv = \frac{2}{4}

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

Check the solution

True

2 uv = \frac{2}{4}

Plug in

Refine

\frac{2}{4} = \frac{2}{4}

True

Check the solution

True

2 uv = \frac{2}{4}

Plug in

Refine

\frac{2}{4} = \frac{2}{4}

True

Therefore, the final solutions for

u^{2} - v^{2} = 0, 2 uv = \frac{2}{4}

are

Substitute back

w = u + v i

Solve

w^{2} = - i \frac{2}{4}

Substitute

w = u + v i

(u + v i)^{2} = - i \frac{2}{4}

Expand

(u + v i)^{2} : (u^{2} - v^{2}) + 2 i uv

(u + v i)^{2}

= (u + i v)^{2}

Apply Perfect Square Formula:

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

a = u, b = v i

= u^{2} + 2 uv i + (v i)^{2}

(v i)^{2} = - v^{2}

(v i)^{2}

Apply exponent rule:

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

= i^{2} v^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) v^{2}

Refine

= - v^{2}

= u^{2} + 2 i uv - v^{2}

Rewrite

u^{2} + 2 i uv - v^{2}

in standard complex form:

(u^{2} - v^{2}) + 2 uv i

u^{2} + 2 i uv - v^{2}

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

= (u^{2} - v^{2}) + 2 uv i

= (u^{2} - v^{2}) + 2 uv i

(u^{2} - v^{2}) + 2 i uv = - i \frac{2}{4}

Rewrite

- i \frac{2}{4}

in standard complex form:

0 - \frac{2}{4} i

(u^{2} - v^{2}) + 2 i uv = 0 - \frac{2}{4} i

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

[u^{2} - v^{2} = 0 2 uv = - \frac{2}{4}]

[u^{2} - v^{2} = 0 2 uv = - \frac{2}{4}]

Isolate

u

for

2 uv = - \frac{2}{4} : u = - \frac{1}{2 ^{\frac{5}{2}} v}

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

Factor the number:

4 = 2 \cdot 2

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

Apply radical rule:

a = a a

2 = 22

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

Cancel the common factor:

2

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

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

Divide both sides by

2 v

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

Divide both sides by

2 v

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

Simplify

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

Simplify

\frac{2 uv}{2 v} : u

\frac{2 uv}{2 v}

Cancel the common factor:

2

= \frac{uv}{v}

Cancel the common factor:

v

= u

Simplify

\frac{- \frac{1}{2 2}}{2 v} : - \frac{1}{2 ^{\frac{5}{2}} v}

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Simplify

22 \cdot 2 v : 2^{\frac{5}{2}} v

22 \cdot 2 v

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} 2 v

Apply radical rule:

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

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

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

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}} v

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

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

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

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

Plug the solutions

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

into

u^{2} - v^{2} = 0

For

u^{2} - v^{2} = 0

, subsitute

u

with

For

u^{2} - v^{2} = 0

, subsitute

u

with

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

(- \frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2} = 0

Solve

(- \frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2} = 0

Simplify

(- \frac{1}{2 ^{\frac{5}{2}} v})^{2} : \frac{1}{32 v ^{2}}

(- \frac{1}{2 ^{\frac{5}{2}} v})^{2}

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

\frac{1}{2 ^{\frac{5}{2}} v}

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

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 2^{2} 2

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

= (- \frac{1}{2 ^{2} 2 v})^{2}

Apply exponent rule:

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

n

is even

(- \frac{1}{2 ^{2} 2 v})^{2} = (\frac{1}{2 ^{2} 2 v})^{2}

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

Apply exponent rule:

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

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

Apply exponent rule:

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

(2^{2} 2 v)^{2} = (2^{2})^{2} (2)^{2} v^{2}

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

(2^{2})^{2} : 2^{4}

Apply exponent rule:

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

= 2^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 2^{4}

= \frac{1 ^{2}}{2 ^{4} ( 2 ) ^{2} v ^{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

= \frac{1 ^{2}}{2 ^{4} \cdot 2 v ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{4} \cdot 2 v ^{2}}

2^{4} \cdot 2 v^{2} = 2^{5} v^{2}

2^{4} \cdot 2 v^{2}

Apply exponent rule:

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

2^{4} \cdot 2 = 2^{4 + 1}

= 2^{4 + 1} v^{2}

Add the numbers:

4 + 1 = 5

= 2^{5} v^{2}

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

2^{5} = 32

= \frac{1}{32 v ^{2}}

\frac{1}{32 v ^{2}} - v^{2} = 0

Multiply both sides by

32 v^{2}

\frac{1}{32 v ^{2}} - v^{2} = 0

Multiply both sides by

32 v^{2}

\frac{1}{32 v ^{2}} \cdot 32 v^{2} - v^{2} \cdot 32 v^{2} = 0 \cdot 32 v^{2}

Simplify

\frac{1}{32 v ^{2}} \cdot 32 v^{2} - v^{2} \cdot 32 v^{2} = 0 \cdot 32 v^{2}

Simplify

\frac{1}{32 v ^{2}} \cdot 32 v^{2} : 1

\frac{1}{32 v ^{2}} \cdot 32 v^{2}

Multiply fractions:

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

= \frac{1 \cdot 32 v ^{2}}{32 v ^{2}}

Cancel the common factor:

32

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

Cancel the common factor:

v^{2}

= 1

Simplify

- v^{2} \cdot 32 v^{2} : - 32 v^{4}

- v^{2} \cdot 32 v^{2}

Apply exponent rule:

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

v^{2} v^{2} = v^{2 + 2}

= - 32 v^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 32 v^{4}

Simplify

0 \cdot 32 v^{2} : 0

0 \cdot 32 v^{2}

Apply rule

0 \cdot a = 0

= 0

1 - 32 v^{4} = 0

1 - 32 v^{4} = 0

1 - 32 v^{4} = 0

Solve

1 - 32 v^{4} = 0

Move

1

to the right side

1 - 32 v^{4} = 0

Subtract

1

from both sides

1 - 32 v^{4} - 1 = 0 - 1

Simplify

- 32 v^{4} = - 1

- 32 v^{4} = - 1

Divide both sides by

- 32

- 32 v^{4} = - 1

Divide both sides by

- 32

\frac{- 32 v ^{4}}{- 32} = \frac{- 1}{- 32}

Simplify

v^{4} = \frac{1}{32}

v^{4} = \frac{1}{32}

For

x^{n} = f (a)

, n is even, the solutions are

Apply radical rule:

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 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

= 2^{5}

Apply exponent rule:

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

2^{5} = 2^{4} \cdot 2

Apply radical rule:

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 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

= 2^{5}

Apply exponent rule:

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

2^{5} = 2^{4} \cdot 2

Apply radical rule:

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

(- \frac{1}{2 ^{\frac{5}{2}} v})^{2} - v^{2}

and compare to zero

Solve

2^{\frac{5}{2}} v = 0 : v = 0

2^{\frac{5}{2}} v = 0

Divide both sides by

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

2^{\frac{5}{2}} v = 0

Divide both sides by

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

\frac{2 ^{\frac{5}{2}} v}{2 ^{\frac{5}{2}}} = \frac{0}{2 ^{\frac{5}{2}}}

Simplify

\frac{2 ^{\frac{5}{2}} v}{2 ^{\frac{5}{2}}} = \frac{0}{2 ^{\frac{5}{2}}}

Simplify

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

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

Cancel the common factor:

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

= v

Simplify

\frac{0}{2 ^{\frac{5}{2}}} : 0

\frac{0}{2 ^{\frac{5}{2}}}

Apply rule

\frac{0}{a} = 0

a \neq = 0

= 0

v = 0

v = 0

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

Plug the solutions

into

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

For

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

, subsitute

v

with

For

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

, subsitute

v

with

Solve

Factor the number:

4 = 2 \cdot 2

Apply radical rule:

a = a a

2 = 22

Cancel the common factor:

2

Multiply both sides by

Simplify

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}

Apply radical rule:

2^{2} \cdot 2^{\frac{1}{4}} = 2^{\frac{9}{4}}

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

Apply exponent rule:

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

2^{2} \cdot 2^{\frac{1}{4}} = 2^{2 + \frac{1}{4}}

= 2^{2 + \frac{1}{4}}

2 + \frac{1}{4} = \frac{9}{4}

2 + \frac{1}{4}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 4 + 1 = 9

2 \cdot 4 + 1

Multiply the numbers:

2 \cdot 4 = 8

= 8 + 1

Add the numbers:

8 + 1 = 9

= 9

= \frac{9}{4}

= 2^{\frac{9}{4}}

Apply the fraction rule:

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

Apply rule:

a \cdot 1 = a

2^{\frac{9}{4}} u \cdot 1 = 2^{\frac{9}{4}} u

Cancel

Simplify

\frac{2 ^{\frac{9}{4}}}{2} : 2^{\frac{5}{4}}

\frac{2 ^{\frac{9}{4}}}{2}

Apply exponent rule:

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

= 2^{\frac{9}{4} - 1}

\frac{9}{4} - 1 = \frac{5}{4}

\frac{9}{4} - 1

Convert element to fraction:

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

= - \frac{1 \cdot 4}{4} + \frac{9}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 4 + 9}{4}

- 1 \cdot 4 + 9 = 5

- 1 \cdot 4 + 9

Multiply the numbers:

1 \cdot 4 = 4

= - 4 + 9

Add/Subtract the numbers:

- 4 + 9 = 5

= 5

= \frac{5}{4}

= 2^{\frac{5}{4}}

Apply radical rule:

= \frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}}

Simplify

\frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}} : 2

\frac{2 ^{\frac{5}{4}}}{2 ^{\frac{1}{4}}}

Apply exponent rule:

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

= 2^{\frac{5}{4} - \frac{1}{4}}

\frac{5}{4} - \frac{1}{4} = 1

\frac{5}{4} - \frac{1}{4}

Apply rule

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

= \frac{5 - 1}{4}

Subtract the numbers:

5 - 1 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply exponent rule:

a^{1} = a

= 2

= 2

= 2 u

= 2 u

Simplify

Apply rule:

(- a) = - a

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

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

Apply the fraction rule:

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

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

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

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

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2 2}

= \frac{2}{2 2}

Cancel the common factor:

2

= \frac{1}{2}

Divide both sides by

2

Divide both sides by

2

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

Apply radical rule:

Cancel the common factor:

Apply exponent rule:

aaa = a^{3}

Apply the fraction rule:

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

Apply radical rule:

= (2^{\frac{1}{4}})^{3}

Apply exponent rule:

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

= 2^{\frac{1}{4} \cdot 3}

\frac{1}{4} \cdot 3 = \frac{3}{4}

\frac{1}{4} \cdot 3

Multiply fractions:

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

= \frac{1 \cdot 3}{4}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{4}

= 2^{\frac{3}{4}}

= - \frac{\frac{1}{2}}{2 ^{\frac{3}{4}}}

- \frac{\frac{1}{2}}{2 ^{\frac{3}{4}}} = - \frac{1}{2 ^{\frac{5}{4}}}

- \frac{\frac{1}{2}}{2 ^{\frac{3}{4}}}

Apply the fraction rule:

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

\frac{\frac{1}{2}}{2 ^{\frac{3}{4}}} = \frac{1}{2 \cdot 2 ^{\frac{3}{4}}}

= - \frac{1}{2 \cdot 2 ^{\frac{3}{4}}}

2 \cdot 2^{\frac{3}{4}} = 2^{\frac{5}{4}}

2 \cdot 2^{\frac{3}{4}}

Apply radical rule:

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

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

= 2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}

Apply exponent rule:

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

2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}} = 2^{\frac{1}{2} + \frac{3}{4}}

= 2^{\frac{1}{2} + \frac{3}{4}}

\frac{1}{2} + \frac{3}{4} = \frac{5}{4}

\frac{1}{2} + \frac{3}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

= \frac{2}{4} + \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 3}{4}

Add the numbers:

2 + 3 = 5

= \frac{5}{4}

= 2^{\frac{5}{4}}

= - \frac{1}{2 ^{\frac{5}{4}}}

= - \frac{1}{2 ^{\frac{5}{4}}}

= - \frac{1}{2 ^{\frac{5}{4}}}

u = - \frac{1}{2 ^{\frac{5}{4}}}

u = - \frac{1}{2 ^{\frac{5}{4}}}

u = - \frac{1}{2 ^{\frac{5}{4}}}

For

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

, subsitute

v

with

For

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

, subsitute

v

with

Solve

Factor the number:

4 = 2 \cdot 2

Apply radical rule:

a = a a

2 = 22

Cancel the common factor:

2

Divide both sides by

Simplify

Apply rule:

a (- b) = - ab

Apply rule:

a (- b) = - ab

Cancel the common factor:

- 2

Cancel the common factor:

= u

Simplify

Apply the fraction rule:

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

Apply rule:

a (- b) = - ab

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

Apply the fraction rule:

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

Multiply the numbers:

2 \cdot 1 = 2

Multiply the numbers:

1 \cdot 2 = 2

Cancel the common factor:

2

Apply the fraction rule:

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

Apply rule:

- (- a) = a

Apply the fraction rule:

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

Cancel

Apply rule:

1 \cdot a = a

Multiply the numbers:

2 \cdot 1 = 2

Apply radical rule:

Cancel the common factor:

Apply exponent rule:

aaa = a^{3}

Apply radical rule:

= (2^{\frac{1}{4}})^{3}

Apply exponent rule:

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

= 2^{\frac{1}{4} \cdot 3}

\frac{1}{4} \cdot 3 = \frac{3}{4}

\frac{1}{4} \cdot 3

Multiply fractions:

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

= \frac{1 \cdot 3}{4}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{4}

= 2^{\frac{3}{4}}

= \frac{1}{2 ^{\frac{3}{4}} 2}

2^{\frac{3}{4}} 2 = 2^{\frac{5}{4}}

2^{\frac{3}{4}} 2

Apply radical rule:

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

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

= 2^{\frac{3}{4}} \cdot 2^{\frac{1}{2}}

Apply exponent rule:

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

2^{\frac{3}{4}} \cdot 2^{\frac{1}{2}} = 2^{\frac{3}{4} + \frac{1}{2}}

= 2^{\frac{3}{4} + \frac{1}{2}}

\frac{3}{4} + \frac{1}{2} = \frac{5}{4}

\frac{3}{4} + \frac{1}{2}

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

= \frac{3}{4} + \frac{2}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2}{4}

Add the numbers:

3 + 2 = 5

= \frac{5}{4}

= 2^{\frac{5}{4}}

= \frac{1}{2 ^{\frac{5}{4}}}

= \frac{1}{2 ^{\frac{5}{4}}}

= \frac{1}{2 ^{\frac{5}{4}}}

u = \frac{1}{2 ^{\frac{5}{4}}}

u = \frac{1}{2 ^{\frac{5}{4}}}

u = \frac{1}{2 ^{\frac{5}{4}}}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

u^{2} - v^{2} = 0

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

Check the solution

True

u^{2} - v^{2} = 0

Plug in

Refine

0 = 0

True

Check the solution

True

u^{2} - v^{2} = 0

Plug in

Refine

0 = 0

True

Check the solutions by plugging them into

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

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

Check the solution

True

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

Plug in

Refine

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

True

Check the solution

True

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

Plug in

Refine

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

True

Therefore, the final solutions for

u^{2} - v^{2} = 0, 2 uv = - \frac{2}{4}

are

Substitute back

w = u + v i

The solutions are

Substitute back

w = sin (x)

No Solution

Simplify

Multiply

Multiply fractions:

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

Multiply:

1 i = i

Since the denominators are equal, combine the fractions:

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

Rationalize

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}} ( 1 + i )}{4}

= \frac{2 ^{\frac{3}{4}} ( 1 + i )}{4}

Rewrite

\frac{2 ^{\frac{3}{4}} ( 1 + i )}{4}

in standard complex form:

\frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

\frac{2 ^{\frac{3}{4}} ( 1 + i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{\frac{3}{4}} ( 1 + i )}{2 ^{2}}

Cancel

\frac{2 ^{\frac{3}{4}} ( 1 + i )}{2 ^{2}} : \frac{1 + i}{2 ^{\frac{5}{4}}}

\frac{2 ^{\frac{3}{4}} ( 1 + i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{3}{4}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{3}{4}}}

= \frac{1 + i}{2 ^{2 - \frac{3}{4}}}

Subtract the numbers:

2 - \frac{3}{4} = \frac{5}{4}

= \frac{1 + i}{2 ^{\frac{5}{4}}}

= \frac{1 + i}{2 ^{\frac{5}{4}}}

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Apply the fraction rule:

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

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}}}{4}

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}}}{4}

= \frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

= \frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

No Solution

No Solution

Simplify

Multiply

Multiply fractions:

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

Multiply:

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

Cancel

Apply exponent rule:

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

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

Subtract the numbers:

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

Cancel

Apply radical rule:

= \frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{4}}}

Apply exponent rule:

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

\frac{2 ^{\frac{3}{2}}}{2 ^{\frac{1}{4}}} = 2^{\frac{3}{2} - \frac{1}{4}}

= 2^{\frac{3}{2} - \frac{1}{4}}

Subtract the numbers:

\frac{3}{2} - \frac{1}{4} = \frac{5}{4}

= 2^{\frac{5}{4}}

= 2^{\frac{5}{4}}

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Multiply

Multiply fractions:

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

Multiply:

1 i = i

Since the denominators are equal, combine the fractions:

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

Rationalize

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}} ( - 1 - i )}{4}

= \frac{2 ^{\frac{3}{4}} ( - 1 - i )}{4}

Rewrite

\frac{2 ^{\frac{3}{4}} ( - 1 - i )}{4}

in standard complex form:

- \frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{4} i

\frac{2 ^{\frac{3}{4}} ( - 1 - i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{\frac{3}{4}} ( - 1 - i )}{2 ^{2}}

Cancel

\frac{2 ^{\frac{3}{4}} ( - 1 - i )}{2 ^{2}} : \frac{- 1 - i}{2 ^{\frac{5}{4}}}

\frac{2 ^{\frac{3}{4}} ( - 1 - i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{3}{4}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{3}{4}}}

= \frac{- 1 - i}{2 ^{2 - \frac{3}{4}}}

Subtract the numbers:

2 - \frac{3}{4} = \frac{5}{4}

= \frac{- 1 - i}{2 ^{\frac{5}{4}}}

= \frac{- 1 - i}{2 ^{\frac{5}{4}}}

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Apply the fraction rule:

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

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{3}{4}}}{4}

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{3}{4}}}{4}

= - \frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{4} i

= - \frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{4} i

No Solution

No Solution

Simplify

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Multiply

Multiply fractions:

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

Multiply:

1 i = i

Since the denominators are equal, combine the fractions:

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

Rationalize

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}} ( - 1 + i )}{4}

= \frac{2 ^{\frac{3}{4}} ( - 1 + i )}{4}

Rewrite

\frac{2 ^{\frac{3}{4}} ( - 1 + i )}{4}

in standard complex form:

- \frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

\frac{2 ^{\frac{3}{4}} ( - 1 + i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{\frac{3}{4}} ( - 1 + i )}{2 ^{2}}

Cancel

\frac{2 ^{\frac{3}{4}} ( - 1 + i )}{2 ^{2}} : \frac{- 1 + i}{2 ^{\frac{5}{4}}}

\frac{2 ^{\frac{3}{4}} ( - 1 + i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{3}{4}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{3}{4}}}

= \frac{- 1 + i}{2 ^{2 - \frac{3}{4}}}

Subtract the numbers:

2 - \frac{3}{4} = \frac{5}{4}

= \frac{- 1 + i}{2 ^{\frac{5}{4}}}

= \frac{- 1 + i}{2 ^{\frac{5}{4}}}

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Apply the fraction rule:

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

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}}}{4}

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{3}{4}}}{4}

= - \frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

= - \frac{2 ^{\frac{3}{4}}}{4} + \frac{2 ^{\frac{3}{4}}}{4} i

No Solution

No Solution

Simplify

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Multiply

Multiply fractions:

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

Multiply:

1 i = i

Since the denominators are equal, combine the fractions:

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

Rationalize

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}} ( 1 - i )}{4}

= \frac{2 ^{\frac{3}{4}} ( 1 - i )}{4}

Rewrite

\frac{2 ^{\frac{3}{4}} ( 1 - i )}{4}

in standard complex form:

\frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{4} i

\frac{2 ^{\frac{3}{4}} ( 1 - i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{\frac{3}{4}} ( 1 - i )}{2 ^{2}}

Cancel

\frac{2 ^{\frac{3}{4}} ( 1 - i )}{2 ^{2}} : \frac{1 - i}{2 ^{\frac{5}{4}}}

\frac{2 ^{\frac{3}{4}} ( 1 - i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{3}{4}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{3}{4}}}

= \frac{1 - i}{2 ^{2 - \frac{3}{4}}}

Subtract the numbers:

2 - \frac{3}{4} = \frac{5}{4}

= \frac{1 - i}{2 ^{\frac{5}{4}}}

= \frac{1 - i}{2 ^{\frac{5}{4}}}

2^{\frac{5}{4}}

2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}}

= 2^{1 + \frac{1}{4}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

Apply the fraction rule:

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

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{3}{4}}}{4}

Multiply by the conjugate

\frac{2 ^{\frac{3}{4}}}{2 ^{\frac{3}{4}}}

1 \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4}}

Apply exponent rule:

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

= 2^{1 + \frac{3}{4} + \frac{1}{4}}

Join

1 + \frac{3}{4} + \frac{1}{4} : 2

1 + \frac{3}{4} + \frac{1}{4}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{3}{4} + \frac{1}{4}

Least Common Multiplier of

1, 4, 4 : 4

1, 4, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{1} :

multiply the denominator and numerator by

4

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

= \frac{4}{4} + \frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 3 + 1}{4}

Add the numbers:

4 + 3 + 1 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{3}{4}}}{4}

= \frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{4} i

= \frac{2 ^{\frac{3}{4}}}{4} - \frac{2 ^{\frac{3}{4}}}{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 sin^4(x)=-1/8 ?
The general solution for sin^4(x)=-1/8 is No Solution for x\in\mathbb{R}