Basic Math Examples

${(\frac{- 16 a^{5} b^{9}}{8 a^{7} b^{- 6}})}^{3}$

Step 1

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

${(\frac{- 16 a^{5} b^{9} b^{6}}{8 a^{7}})}^{3}$

Step 2

Multiply $b^{9}$ by $b^{6}$ by adding the exponents.

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Step 2.1

Move $b^{6}$ .

${(\frac{- 16 a^{5} (b^{6} b^{9})}{8 a^{7}})}^{3}$

Step 2.2

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

${(\frac{- 16 a^{5} b^{6 + 9}}{8 a^{7}})}^{3}$

Step 2.3

Add $6$ and $9$ .

${(\frac{- 16 a^{5} b^{15}}{8 a^{7}})}^{3}$

Step 3

Cancel the common factor of $- 16$ and $8$ .

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Step 3.1

Factor $8$ out of $- 16 a^{5} b^{15}$ .

${(\frac{8 (- 2 a^{5} b^{15})}{8 a^{7}})}^{3}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $8$ out of $8 a^{7}$ .

${(\frac{8 (- 2 a^{5} b^{15})}{8 (a^{7})})}^{3}$

Step 3.2.2

Cancel the common factor.

${(\frac{8 (- 2 a^{5} b^{15})}{8 a^{7}})}^{3}$

Step 3.2.3

Rewrite the expression.

${(\frac{- 2 a^{5} b^{15}}{a^{7}})}^{3}$

Step 4

Cancel the common factor of $a^{5}$ and $a^{7}$ .

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Step 4.1

Factor $a^{5}$ out of $- 2 a^{5} b^{15}$ .

${(\frac{a^{5} (- 2 b^{15})}{a^{7}})}^{3}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $a^{5}$ out of $a^{7}$ .

${(\frac{a^{5} (- 2 b^{15})}{a^{5} a^{2}})}^{3}$

Step 4.2.2

Cancel the common factor.

${(\frac{a^{5} (- 2 b^{15})}{a^{5} a^{2}})}^{3}$

Step 4.2.3

Rewrite the expression.

${(\frac{- 2 b^{15}}{a^{2}})}^{3}$

Step 5

Move the negative in front of the fraction.

${(- \frac{2 b^{15}}{a^{2}})}^{3}$

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 $- \frac{2 b^{15}}{a^{2}}$ .

${(- 1)}^{3} {(\frac{2 b^{15}}{a^{2}})}^{3}$

Step 6.2

Apply the product rule to $\frac{2 b^{15}}{a^{2}}$ .

${(- 1)}^{3} \frac{{(2 b^{15})}^{3}}{{(a^{2})}^{3}}$

Step 6.3

Apply the product rule to $2 b^{15}$ .

${(- 1)}^{3} \frac{2^{3} {(b^{15})}^{3}}{{(a^{2})}^{3}}$

Step 7

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

$- \frac{2^{3} {(b^{15})}^{3}}{{(a^{2})}^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

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

$- \frac{8 {(b^{15})}^{3}}{{(a^{2})}^{3}}$

Step 8.2

Multiply the exponents in ${(b^{15})}^{3}$ .

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Step 8.2.1

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

$- \frac{8 b^{15 \cdot 3}}{{(a^{2})}^{3}}$

Step 8.2.2

Multiply $15$ by $3$ .

$- \frac{8 b^{45}}{{(a^{2})}^{3}}$

Step 9

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

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Step 9.1

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

$- \frac{8 b^{45}}{a^{2 \cdot 3}}$

Step 9.2

Multiply $2$ by $3$ .

$- \frac{8 b^{45}}{a^{6}}$