Algebra Examples

${(a^{3} b^{4})}^{- 2} {(a^{- 3} b^{- 5})}^{- 4}$

Step 1

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

$\frac{1}{{(a^{3} b^{4})}^{2}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $a^{3} b^{4}$ .

$\frac{1}{{(a^{3})}^{2} {(b^{4})}^{2}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{a^{3 \cdot 2} {(b^{4})}^{2}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 2.2.2

Multiply $3$ by $2$ .

$\frac{1}{a^{6} {(b^{4})}^{2}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 2.3

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

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Step 2.3.1

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

$\frac{1}{a^{6} b^{4 \cdot 2}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 2.3.2

Multiply $4$ by $2$ .

$\frac{1}{a^{6} b^{8}} {(a^{- 3} b^{- 5})}^{- 4}$

Step 3

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

$\frac{1}{a^{6} b^{8}} {(\frac{1}{a^{3}} b^{- 5})}^{- 4}$

Step 4

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

$\frac{1}{a^{6} b^{8}} {(\frac{1}{a^{3}} \cdot \frac{1}{b^{5}})}^{- 4}$

Step 5

Combine.

$\frac{1}{a^{6} b^{8}} {(\frac{1 \cdot 1}{a^{3} b^{5}})}^{- 4}$

Step 6

Multiply $1$ by $1$ .

$\frac{1}{a^{6} b^{8}} {(\frac{1}{a^{3} b^{5}})}^{- 4}$

Step 7

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{1}{a^{6} b^{8}} {(a^{3} b^{5})}^{4}$

Step 8

Apply basic rules of exponents.

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Step 8.1

Apply the product rule to $a^{3} b^{5}$ .

$\frac{1}{a^{6} b^{8}} ({(a^{3})}^{4} {(b^{5})}^{4})$

Step 8.2

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

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Step 8.2.1

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

$\frac{1}{a^{6} b^{8}} (a^{3 \cdot 4} {(b^{5})}^{4})$

Step 8.2.2

Multiply $3$ by $4$ .

$\frac{1}{a^{6} b^{8}} (a^{12} {(b^{5})}^{4})$

Step 9

Cancel the common factor of $a^{6}$ .

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Step 9.1

Factor $a^{6}$ out of $a^{6} b^{8}$ .

$\frac{1}{a^{6} (b^{8})} (a^{12} {(b^{5})}^{4})$

Step 9.2

Factor $a^{6}$ out of $a^{12} {(b^{5})}^{4}$ .

$\frac{1}{a^{6} (b^{8})} (a^{6} (a^{6} {(b^{5})}^{4}))$

Step 9.3

Cancel the common factor.

$\frac{1}{a^{6} b^{8}} (a^{6} (a^{6} {(b^{5})}^{4}))$

Step 9.4

Rewrite the expression.

$\frac{1}{b^{8}} (a^{6} {(b^{5})}^{4})$

Step 10

Combine $a^{6}$ and $\frac{1}{b^{8}}$ .

$\frac{a^{6}}{b^{8}} {(b^{5})}^{4}$

Step 11

Combine $\frac{a^{6}}{b^{8}}$ and ${(b^{5})}^{4}$ .

$\frac{a^{6} {(b^{5})}^{4}}{b^{8}}$

Step 12

Multiply the exponents in ${(b^{5})}^{4}$ .

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Step 12.1

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

$\frac{a^{6} b^{5 \cdot 4}}{b^{8}}$

Step 12.2

Multiply $5$ by $4$ .

$\frac{a^{6} b^{20}}{b^{8}}$

Step 13

Cancel the common factor of $b^{20}$ and $b^{8}$ .

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Step 13.1

Factor $b^{8}$ out of $a^{6} b^{20}$ .

$\frac{b^{8} (a^{6} b^{12})}{b^{8}}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Multiply by $1$ .

$\frac{b^{8} (a^{6} b^{12})}{b^{8} \cdot 1}$

Step 13.2.2

Cancel the common factor.

$\frac{b^{8} (a^{6} b^{12})}{b^{8} \cdot 1}$

Step 13.2.3

Rewrite the expression.

$\frac{a^{6} b^{12}}{1}$

Step 13.2.4

Divide $a^{6} b^{12}$ by $1$ .

$a^{6} b^{12}$