Precalculus Examples

${(\frac{x^{12} y^{- 3}}{8 y^{\frac{3}{2}}})}^{- \frac{1}{3}}$

Step 1

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{x^{12}}{8 y^{\frac{3}{2}} y^{3}})}^{- \frac{1}{3}}$

Step 2

Multiply $y^{\frac{3}{2}}$ by $y^{3}$ by adding the exponents.

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Step 2.1

Move $y^{3}$ .

${(\frac{x^{12}}{8 (y^{3} y^{\frac{3}{2}})})}^{- \frac{1}{3}}$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{x^{12}}{8 y^{3 + \frac{3}{2}}})}^{- \frac{1}{3}}$

Step 2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{x^{12}}{8 y^{3 \cdot \frac{2}{2} + \frac{3}{2}}})}^{- \frac{1}{3}}$

Step 2.4

Combine $3$ and $\frac{2}{2}$ .

${(\frac{x^{12}}{8 y^{\frac{3 \cdot 2}{2} + \frac{3}{2}}})}^{- \frac{1}{3}}$

Step 2.5

Combine the numerators over the common denominator.

${(\frac{x^{12}}{8 y^{\frac{3 \cdot 2 + 3}{2}}})}^{- \frac{1}{3}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $3$ by $2$ .

${(\frac{x^{12}}{8 y^{\frac{6 + 3}{2}}})}^{- \frac{1}{3}}$

Step 2.6.2

Add $6$ and $3$ .

${(\frac{x^{12}}{8 y^{\frac{9}{2}}})}^{- \frac{1}{3}}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{8 y^{\frac{9}{2}}}{x^{12}})}^{\frac{1}{3}}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{8 y^{\frac{9}{2}}}{x^{12}}$ .

$\frac{{(8 y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 4.2

Apply the product rule to $8 y^{\frac{9}{2}}$ .

$\frac{8^{\frac{1}{3}} {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $8$ as $2^{3}$ .

$\frac{{(2^{3})}^{\frac{1}{3}} {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{3 (\frac{1}{3})} {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Cancel the common factor.

$\frac{2^{3 (\frac{1}{3})} {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.3.2

Rewrite the expression.

$\frac{2^{1} {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.4

Evaluate the exponent.

$\frac{2 {(y^{\frac{9}{2}})}^{\frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.5

Multiply the exponents in ${(y^{\frac{9}{2}})}^{\frac{1}{3}}$ .

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Step 5.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 y^{\frac{9}{2} \cdot \frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.5.2

Cancel the common factor of $3$ .

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Step 5.5.2.1

Factor $3$ out of $9$ .

$\frac{2 y^{\frac{3 (3)}{2} \cdot \frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.5.2.2

Cancel the common factor.

$\frac{2 y^{\frac{3 \cdot 3}{2} \cdot \frac{1}{3}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 5.5.2.3

Rewrite the expression.

$\frac{2 y^{\frac{3}{2}}}{{(x^{12})}^{\frac{1}{3}}}$

Step 6

Multiply the exponents in ${(x^{12})}^{\frac{1}{3}}$ .

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Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 y^{\frac{3}{2}}}{x^{12 (\frac{1}{3})}}$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Factor $3$ out of $12$ .

$\frac{2 y^{\frac{3}{2}}}{x^{3 (4) \frac{1}{3}}}$

Step 6.2.2

Cancel the common factor.

$\frac{2 y^{\frac{3}{2}}}{x^{3 \cdot 4 \frac{1}{3}}}$

Step 6.2.3

Rewrite the expression.

$\frac{2 y^{\frac{3}{2}}}{x^{4}}$