Precalculus Examples

${(\frac{- 8 a^{3}}{a^{- 6}})}^{\frac{2}{3}}$

Step 1

Move $a^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(- 8 a^{3} a^{6})}^{\frac{2}{3}}$

Step 2

Multiply $a^{3}$ by $a^{6}$ by adding the exponents.

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Step 2.1

Move $a^{6}$ .

${(- 8 (a^{6} a^{3}))}^{\frac{2}{3}}$

Step 2.2

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

${(- 8 a^{6 + 3})}^{\frac{2}{3}}$

Step 2.3

Add $6$ and $3$ .

${(- 8 a^{9})}^{\frac{2}{3}}$

Step 3

Simplify the expression.

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Step 3.1

Apply the product rule to $- 8 a^{9}$ .

${(- 8)}^{\frac{2}{3}} {(a^{9})}^{\frac{2}{3}}$

Step 3.2

Rewrite $- 8$ as ${(- 2)}^{3}$ .

${({(- 2)}^{3})}^{\frac{2}{3}} {(a^{9})}^{\frac{2}{3}}$

Step 3.3

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

${(- 2)}^{3 (\frac{2}{3})} {(a^{9})}^{\frac{2}{3}}$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Cancel the common factor.

${(- 2)}^{3 (\frac{2}{3})} {(a^{9})}^{\frac{2}{3}}$

Step 4.2

Rewrite the expression.

${(- 2)}^{2} {(a^{9})}^{\frac{2}{3}}$

Step 5

Simplify the expression.

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Step 5.1

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

$4 {(a^{9})}^{\frac{2}{3}}$

Step 5.2

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

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Step 5.2.1

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

$4 a^{9 (\frac{2}{3})}$

Step 5.2.2

Cancel the common factor of $3$ .

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Step 5.2.2.1

Factor $3$ out of $9$ .

$4 a^{3 (3) \frac{2}{3}}$

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

$4 a^{3 \cdot 2}$

Step 5.2.3

Multiply $3$ by $2$ .

$4 a^{6}$