Precalculus Examples

${(27 x^{- 6} y^{9})}^{\frac{2}{3}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(27 \frac{1}{x^{6}} y^{9})}^{\frac{2}{3}}$

Step 2

Combine $27$ and $\frac{1}{x^{6}}$ .

${(\frac{27}{x^{6}} y^{9})}^{\frac{2}{3}}$

Step 3

Combine $\frac{27}{x^{6}}$ and $y^{9}$ .

${(\frac{27 y^{9}}{x^{6}})}^{\frac{2}{3}}$

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 $\frac{27 y^{9}}{x^{6}}$ .

$\frac{{(27 y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 4.2

Apply the product rule to $27 y^{9}$ .

$\frac{27^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $27$ as $3^{3}$ .

$\frac{{(3^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.2

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

$\frac{3^{3 (\frac{2}{3})} {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{2}{3})} {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.3.2

Rewrite the expression.

$\frac{3^{2} {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.4

Raise $3$ to the power of $2$ .

$\frac{9 {(y^{9})}^{\frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.5

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

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Step 5.5.1

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

$\frac{9 y^{9 (\frac{2}{3})}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.5.2

Cancel the common factor of $3$ .

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Step 5.5.2.1

Factor $3$ out of $9$ .

$\frac{9 y^{3 (3) \frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.5.2.2

Cancel the common factor.

$\frac{9 y^{3 \cdot 3 \frac{2}{3}}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.5.2.3

Rewrite the expression.

$\frac{9 y^{3 \cdot 2}}{{(x^{6})}^{\frac{2}{3}}}$

Step 5.5.3

Multiply $3$ by $2$ .

$\frac{9 y^{6}}{{(x^{6})}^{\frac{2}{3}}}$

Step 6

Multiply the exponents in ${(x^{6})}^{\frac{2}{3}}$ .

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Step 6.1

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

$\frac{9 y^{6}}{x^{6 (\frac{2}{3})}}$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Factor $3$ out of $6$ .

$\frac{9 y^{6}}{x^{3 (2) \frac{2}{3}}}$

Step 6.2.2

Cancel the common factor.

$\frac{9 y^{6}}{x^{3 \cdot 2 \frac{2}{3}}}$

Step 6.2.3

Rewrite the expression.

$\frac{9 y^{6}}{x^{2 \cdot 2}}$

Step 6.3

Multiply $2$ by $2$ .

$\frac{9 y^{6}}{x^{4}}$