Finite Math Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $2 a^{2} b^{- 1}$ .

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

Step 2.2

Apply the product rule to $2 a^{2}$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $3$ .

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

Step 2.5

Multiply the exponents in ${(b^{- 1})}^{3}$ .

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Step 2.5.1

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

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

Step 2.5.2

Multiply $- 1$ by $3$ .

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

Step 2.6

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

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

Step 2.7

Combine exponents.

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Step 2.7.1

Combine $8$ and $\frac{1}{b^{3}}$ .

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

Step 2.7.2

Combine $a^{6}$ and $\frac{8}{b^{3}}$ .

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

Step 2.8

Move $8$ to the left of $a^{6}$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $3 a^{- 1} b^{2}$ .

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

Step 3.2

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

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

Step 3.3

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

$\frac{\frac{8 a^{6}}{b^{3}}}{{(\frac{3}{a})}^{2} {(b^{2})}^{2}}$

Step 3.4

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

$\frac{\frac{8 a^{6}}{b^{3}}}{\frac{3^{2}}{a^{2}} {(b^{2})}^{2}}$

Step 3.5

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

$\frac{\frac{8 a^{6}}{b^{3}}}{\frac{9}{a^{2}} {(b^{2})}^{2}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{\frac{8 a^{6}}{b^{3}}}{\frac{9}{a^{2}} b^{2 \cdot 2}}$

Step 3.6.2

Multiply $2$ by $2$ .

$\frac{\frac{8 a^{6}}{b^{3}}}{\frac{9}{a^{2}} b^{4}}$

Step 4

Combine $\frac{9}{a^{2}}$ and $b^{4}$ .

$\frac{\frac{8 a^{6}}{b^{3}}}{\frac{9 b^{4}}{a^{2}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 a^{6}}{b^{3}} \cdot \frac{a^{2}}{9 b^{4}}$

Step 6

Combine.

$\frac{8 a^{6} a^{2}}{b^{3} (9 b^{4})}$

Step 7

Multiply $a^{6}$ by $a^{2}$ by adding the exponents.

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Step 7.1

Move $a^{2}$ .

$\frac{8 (a^{2} a^{6})}{b^{3} (9 b^{4})}$

Step 7.2

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

$\frac{8 a^{2 + 6}}{b^{3} (9 b^{4})}$

Step 7.3

Add $2$ and $6$ .

$\frac{8 a^{8}}{b^{3} (9 b^{4})}$

Step 8

Multiply $b^{3}$ by $b^{4}$ by adding the exponents.

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Step 8.1

Move $b^{4}$ .

$\frac{8 a^{8}}{b^{4} b^{3} \cdot 9}$

Step 8.2

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

$\frac{8 a^{8}}{b^{4 + 3} \cdot 9}$

Step 8.3

Add $4$ and $3$ .

$\frac{8 a^{8}}{b^{7} \cdot 9}$

Step 9

Move $9$ to the left of $b^{7}$ .

$\frac{8 a^{8}}{9 b^{7}}$