Algebra Examples

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

Step 1

Move ${(3 a^{5} b)}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

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

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

Step 3

Combine $2$ and $\frac{1}{a^{3}}$ .

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

Step 4

Combine $\frac{2}{a^{3}}$ and $b^{4}$ .

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

Step 5

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

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Step 5.1

Apply the product rule to $\frac{2 b^{4}}{a^{3}}$ .

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

Step 5.2

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

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $4$ by $2$ .

${(\frac{4 b^{8}}{{(a^{3})}^{2}} {(3 a^{5} b)}^{2})}^{- 1}$

Step 7

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

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Step 7.1

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

${(\frac{4 b^{8}}{a^{3 \cdot 2}} {(3 a^{5} b)}^{2})}^{- 1}$

Step 7.2

Multiply $3$ by $2$ .

${(\frac{4 b^{8}}{a^{6}} {(3 a^{5} b)}^{2})}^{- 1}$

Step 8

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

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Step 8.1

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

${(\frac{4 b^{8}}{a^{6}} ({(3 a^{5})}^{2} b^{2}))}^{- 1}$

Step 8.2

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

${(\frac{4 b^{8}}{a^{6}} (3^{2} {(a^{5})}^{2} b^{2}))}^{- 1}$

Step 9

Rewrite using the commutative property of multiplication.

${(3^{2} \frac{4 b^{8}}{a^{6}} ({(a^{5})}^{2} b^{2}))}^{- 1}$

Step 10

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

${(9 \frac{4 b^{8}}{a^{6}} ({(a^{5})}^{2} b^{2}))}^{- 1}$

Step 11

Multiply $9 \frac{4 b^{8}}{a^{6}}$ .

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Step 11.1

Combine $9$ and $\frac{4 b^{8}}{a^{6}}$ .

${(\frac{9 (4 b^{8})}{a^{6}} ({(a^{5})}^{2} b^{2}))}^{- 1}$

Step 11.2

Multiply $4$ by $9$ .

${(\frac{36 b^{8}}{a^{6}} ({(a^{5})}^{2} b^{2}))}^{- 1}$

Step 12

Multiply the exponents in ${(a^{5})}^{2}$ .

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Step 12.1

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

${(\frac{36 b^{8}}{a^{6}} (a^{5 \cdot 2} b^{2}))}^{- 1}$

Step 12.2

Multiply $5$ by $2$ .

${(\frac{36 b^{8}}{a^{6}} (a^{10} b^{2}))}^{- 1}$

Step 13

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

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Step 13.1

Factor $a^{6}$ out of $a^{10} b^{2}$ .

${(\frac{36 b^{8}}{a^{6}} (a^{6} (a^{4} b^{2})))}^{- 1}$

Step 13.2

Cancel the common factor.

${(\frac{36 b^{8}}{a^{6}} (a^{6} (a^{4} b^{2})))}^{- 1}$

Step 13.3

Rewrite the expression.

${(36 b^{8} (a^{4} b^{2}))}^{- 1}$

Step 14

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

${(36 (a^{4} b^{8 + 2}))}^{- 1}$

Step 15

Add $8$ and $2$ .

${(36 (a^{4} b^{10}))}^{- 1}$

Step 16

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

$\frac{1}{36 a^{4} b^{10}}$