Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

$\frac{9 a^{4 \cdot 2} {(b^{- 2})}^{2}}{{(2 a^{- 3} b)}^{3}}$

Step 1.4.2

Multiply $4$ by $2$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Multiply $- 2$ by $2$ .

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

Step 1.6

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

$\frac{9 a^{8} \frac{1}{b^{4}}}{{(2 a^{- 3} b)}^{3}}$

Step 1.7

Combine exponents.

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Step 1.7.1

Combine $9$ and $\frac{1}{b^{4}}$ .

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

Step 1.7.2

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

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

Step 1.8

Move $9$ to the left of $a^{8}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

$\frac{\frac{9 a^{8}}{b^{4}}}{{(2 \frac{1}{a^{3}})}^{3} b^{3}}$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Multiply $3$ by $3$ .

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

Step 3

Combine $\frac{8}{a^{9}}$ and $b^{3}$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply $a^{8}$ by $a^{9}$ by adding the exponents.

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Step 6.1

Move $a^{9}$ .

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

Step 6.2

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

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

Step 6.3

Add $9$ and $8$ .

$\frac{9 a^{17}}{b^{4} (8 b^{3})}$

Step 7

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

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Step 7.1

Move $b^{3}$ .

$\frac{9 a^{17}}{b^{3} b^{4} \cdot 8}$

Step 7.2

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

$\frac{9 a^{17}}{b^{3 + 4} \cdot 8}$

Step 7.3

Add $3$ and $4$ .

$\frac{9 a^{17}}{b^{7} \cdot 8}$

Step 8

Move $8$ to the left of $b^{7}$ .

$\frac{9 a^{17}}{8 b^{7}}$