Precalculus Examples

${(\frac{- 4 a^{14} b^{8}}{12 a^{17} b^{- 2}})}^{3}$

Step 1

Move $b^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{- 4 a^{14} b^{8} b^{2}}{12 a^{17}})}^{3}$

Step 2

Multiply $b^{8}$ by $b^{2}$ by adding the exponents.

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Step 2.1

Move $b^{2}$ .

${(\frac{- 4 a^{14} (b^{2} b^{8})}{12 a^{17}})}^{3}$

Step 2.2

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

${(\frac{- 4 a^{14} b^{2 + 8}}{12 a^{17}})}^{3}$

Step 2.3

Add $2$ and $8$ .

${(\frac{- 4 a^{14} b^{10}}{12 a^{17}})}^{3}$

Step 3

Cancel the common factor of $- 4$ and $12$ .

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Step 3.1

Factor $4$ out of $- 4 a^{14} b^{10}$ .

${(\frac{4 (- a^{14} b^{10})}{12 a^{17}})}^{3}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $12 a^{17}$ .

${(\frac{4 (- a^{14} b^{10})}{4 (3 a^{17})})}^{3}$

Step 3.2.2

Cancel the common factor.

${(\frac{4 (- a^{14} b^{10})}{4 (3 a^{17})})}^{3}$

Step 3.2.3

Rewrite the expression.

${(\frac{- a^{14} b^{10}}{3 a^{17}})}^{3}$

Step 4

Cancel the common factor of $a^{14}$ and $a^{17}$ .

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Step 4.1

Factor $a^{14}$ out of $- a^{14} b^{10}$ .

${(\frac{a^{14} (- 1 b^{10})}{3 a^{17}})}^{3}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $a^{14}$ out of $3 a^{17}$ .

${(\frac{a^{14} (- 1 b^{10})}{a^{14} (3 a^{3})})}^{3}$

Step 4.2.2

Cancel the common factor.

${(\frac{a^{14} (- 1 b^{10})}{a^{14} (3 a^{3})})}^{3}$

Step 4.2.3

Rewrite the expression.

${(\frac{- 1 b^{10}}{3 a^{3}})}^{3}$

Step 5

Move the negative in front of the fraction.

${(- \frac{b^{10}}{3 a^{3}})}^{3}$

Step 6

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

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Step 6.1

Apply the product rule to $- \frac{b^{10}}{3 a^{3}}$ .

${(- 1)}^{3} {(\frac{b^{10}}{3 a^{3}})}^{3}$

Step 6.2

Apply the product rule to $\frac{b^{10}}{3 a^{3}}$ .

${(- 1)}^{3} \frac{{(b^{10})}^{3}}{{(3 a^{3})}^{3}}$

Step 6.3

Apply the product rule to $3 a^{3}$ .

${(- 1)}^{3} \frac{{(b^{10})}^{3}}{3^{3} {(a^{3})}^{3}}$

Step 7

Simplify the expression.

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Step 7.1

Raise $- 1$ to the power of $3$ .

$- \frac{{(b^{10})}^{3}}{3^{3} {(a^{3})}^{3}}$

Step 7.2

Multiply the exponents in ${(b^{10})}^{3}$ .

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Step 7.2.1

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

$- \frac{b^{10 \cdot 3}}{3^{3} {(a^{3})}^{3}}$

Step 7.2.2

Multiply $10$ by $3$ .

$- \frac{b^{30}}{3^{3} {(a^{3})}^{3}}$

Step 8

Simplify the denominator.

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Step 8.1

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

$- \frac{b^{30}}{27 {(a^{3})}^{3}}$

Step 8.2

Multiply the exponents in ${(a^{3})}^{3}$ .

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Step 8.2.1

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

$- \frac{b^{30}}{27 a^{3 \cdot 3}}$

Step 8.2.2

Multiply $3$ by $3$ .

$- \frac{b^{30}}{27 a^{9}}$