Algebra Examples

$\frac{{(4 a^{- 1} b^{- 4})}^{- 3}}{{(- 8 a^{4} b^{- 2})}^{- 2}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(4 a^{- 1} b^{- 4})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(- 8 a^{4} b^{- 2})}^{- 2} {(4 a^{- 1} b^{- 4})}^{3}}$

Step 1.2

Move ${(- 8 a^{4} b^{- 2})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(- 8 a^{4} b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $- 8 a^{4} b^{- 2}$ .

$\frac{{(- 8 a^{4})}^{2} {(b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.2

Apply the product rule to $- 8 a^{4}$ .

$\frac{{(- 8)}^{2} {(a^{4})}^{2} {(b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.3

Raise $- 8$ to the power of $2$ .

$\frac{64 {(a^{4})}^{2} {(b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.4

Multiply the exponents in ${(a^{4})}^{2}$ .

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Step 2.4.1

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

$\frac{64 a^{4 \cdot 2} {(b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.4.2

Multiply $4$ by $2$ .

$\frac{64 a^{8} {(b^{- 2})}^{2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.5

Multiply the exponents in ${(b^{- 2})}^{2}$ .

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Step 2.5.1

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

$\frac{64 a^{8} b^{- 2 \cdot 2}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.5.2

Multiply $- 2$ by $2$ .

$\frac{64 a^{8} b^{- 4}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.6

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

$\frac{64 a^{8} \frac{1}{b^{4}}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.7

Combine exponents.

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Step 2.7.1

Combine $64$ and $\frac{1}{b^{4}}$ .

$\frac{a^{8} \frac{64}{b^{4}}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.7.2

Combine $a^{8}$ and $\frac{64}{b^{4}}$ .

$\frac{\frac{a^{8} \cdot 64}{b^{4}}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 2.8

Move $64$ to the left of $a^{8}$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{{(4 a^{- 1} b^{- 4})}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $4 a^{- 1} b^{- 4}$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{{(4 a^{- 1})}^{3} {(b^{- 4})}^{3}}$

Step 3.2

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

$\frac{\frac{64 a^{8}}{b^{4}}}{{(4 \frac{1}{a})}^{3} {(b^{- 4})}^{3}}$

Step 3.3

Combine $4$ and $\frac{1}{a}$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{{(\frac{4}{a})}^{3} {(b^{- 4})}^{3}}$

Step 3.4

Apply the product rule to $\frac{4}{a}$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{4^{3}}{a^{3}} {(b^{- 4})}^{3}}$

Step 3.5

Raise $4$ to the power of $3$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{64}{a^{3}} {(b^{- 4})}^{3}}$

Step 3.6

Multiply the exponents in ${(b^{- 4})}^{3}$ .

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Step 3.6.1

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

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{64}{a^{3}} b^{- 4 \cdot 3}}$

Step 3.6.2

Multiply $- 4$ by $3$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{64}{a^{3}} b^{- 12}}$

Step 3.7

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

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{64}{a^{3}} \cdot \frac{1}{b^{12}}}$

Step 4

Multiply $\frac{64}{a^{3}}$ by $\frac{1}{b^{12}}$ .

$\frac{\frac{64 a^{8}}{b^{4}}}{\frac{64}{a^{3} b^{12}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{64 a^{8}}{b^{4}} \cdot \frac{a^{3} b^{12}}{64}$

Step 6

Combine.

$\frac{64 a^{8} (a^{3} b^{12})}{b^{4} \cdot 64}$

Step 7

Multiply $a^{8}$ by $a^{3}$ by adding the exponents.

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Step 7.1

Move $a^{3}$ .

$\frac{64 (a^{3} a^{8}) b^{12}}{b^{4} \cdot 64}$

Step 7.2

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

$\frac{64 a^{3 + 8} b^{12}}{b^{4} \cdot 64}$

Step 7.3

Add $3$ and $8$ .

$\frac{64 a^{11} b^{12}}{b^{4} \cdot 64}$

Step 8

Cancel the common factor of $64$ .

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Step 8.1

Cancel the common factor.

$\frac{64 a^{11} b^{12}}{b^{4} \cdot 64}$

Step 8.2

Rewrite the expression.

$\frac{a^{11} b^{12}}{b^{4}}$

Step 9

Cancel the common factor of $b^{12}$ and $b^{4}$ .

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Step 9.1

Factor $b^{4}$ out of $a^{11} b^{12}$ .

$\frac{b^{4} (a^{11} b^{8})}{b^{4}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Multiply by $1$ .

$\frac{b^{4} (a^{11} b^{8})}{b^{4} \cdot 1}$

Step 9.2.2

Cancel the common factor.

$\frac{b^{4} (a^{11} b^{8})}{b^{4} \cdot 1}$

Step 9.2.3

Rewrite the expression.

$\frac{a^{11} b^{8}}{1}$

Step 9.2.4

Divide $a^{11} b^{8}$ by $1$ .

$a^{11} b^{8}$