Example

Evaluate [latex] \displaystyle \frac{24{{x}^{8}}}{2{{x}^{5}}}[/latex] when [latex]x=4[/latex].

Answer: Separate into numerical and variable factors. [latex-display] \displaystyle \left( \frac{24}{2} \right)\left( \frac{{{x}^{8}}}{{{x}^{5}}} \right)[/latex-display] Divide coefficients, and subtract the exponents of the variables. [latex-display] \displaystyle 12\left( {{x}^{8-5}} \right)[/latex-display] Simplify. [latex-display] \displaystyle 12{{x}^{3}}[/latex-display] Substitute the value [latex]4[/latex] for the variable [latex]x[/latex]. [latex-display] \displaystyle (12)({{4}^{3}})=12\cdot 64[/latex-display]

Answer

[latex-display] \displaystyle \frac{24{{x}^{8}}}{2{{x}^{5}}}[/latex] = [latex]768[/latex-display]

Example

Evaluate [latex] \displaystyle \frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}[/latex] when [latex]x=4[/latex] and [latex]y=-2[/latex].

Answer: In the denominator, notice that a product is being raised to a power.   Use the rules of exponents to simplify the denominator.

[latex] \displaystyle{\left(2{x}^{3}y\right)}^{2}={2}^{2}{x}^{3\cdot 2}{y}^{2}={2}^{2}{x}^{6}{y}^{2}={4x^{6}y^{2}}[/latex]

Here is the fraction with a simplified denominator:

[latex]\frac{24x^{8}y^{2}}{4x^{6}y^{2}}[/latex]

Separate into numerical and variable factors to simplify further.

[latex] \displaystyle \left( \frac{24}{4} \right)\left( \frac{{{x}^{8}}}{{{x}^{6}}} \right)\left( \frac{{{y}^{2}}}{{{y}^{2}}}\right)[/latex]

Divide coefficients, use the Quotient Rule to divide the variables—subtract the exponents.

[latex] \displaystyle 6\left( {{x}^{8-6}} \right)\left( {{y}^{2-2}} \right)[/latex]

Simplify. Remember that [latex]y^{0}[/latex] is [latex]1[/latex].

[latex] \displaystyle 6{{x}^{2}}{{y}^{0}}=6{{x}^{2}}[/latex]

Substitute the value [latex]4[/latex] for the variable [latex]x[/latex].

[latex] \displaystyle (6)({{4}^{2}})=6\cdot 16[/latex]

Answer

[latex-display] \displaystyle \frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}=96[/latex] when [latex]x=4[/latex] and [latex]y=-2[/latex-display]

Example

Simplify. [latex]a^{2}\left(a^{5}\right)^{3}[/latex]

Answer: Raise [latex]a^{5}[/latex] to the power of [latex]3[/latex] by multiplying the exponents together (the Power Rule).

[latex] \displaystyle {{a}^{2}}{{a}^{5\cdot 3}}[/latex]

Since the exponents share the same base, a, they can be combined (the Product Rule).

[latex] \displaystyle {{a}^{2}}{{a}^{15}}\\{{a}^{2+15}}[/latex]

Answer

[latex-display] \displaystyle {{a}^{2}}{{({{a}^{5}})}^{3}}={{a}^{17}}[/latex-display]

Example

Simplify. [latex] \displaystyle \frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[/latex]

Answer: Use the order of operations. Parentheses, Exponents, Multiply/ Divide, Add/ Subtract There is nothing inside parentheses or brackets that we can simplify further, so we will evaluate exponents first. Use the Power Rule to simplify [latex]\left(a^{5}\right)^{3}[/latex].

[latex]\left(a^{5}\right)^{3}=a^{5\cdot{3}}=a^{15}[/latex]

The expression now looks like this:

[latex] \displaystyle\frac{{{a}^{2}}{{a}^{15}}}{8{{a}^{8}}}[/latex]

Now we can multiply, using the Product Rule to simplify the numerator because the bases are the same.

[latex] \displaystyle{{a}^{2}}{{a}^{15}}=a^{17}[/latex], and the expression looks like this:

[latex]\displaystyle\frac{{{{a}^{17}}}}{{8{{a}^{8}}}}[/latex]

Now we can divide using the Quotient Rule.

[latex] \displaystyle \frac{{{a}^{17-8}}}{8}[/latex]

Answer

[latex-display] \displaystyle \frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}=\frac{{{a}^{9}}}{8}[/latex-display]