Basic Math Examples

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

Step 1

Simplify.

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Step 1.1

Move $b^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $3$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $- 2$ by $3$ .

$\frac{a^{4} a^{9} b^{- 6}}{b^{6}}$

Step 2.4

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

$\frac{a^{4} a^{9} \frac{1}{b^{6}}}{b^{6}}$

Step 2.5

Combine exponents.

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Step 2.5.1

Multiply $a^{4}$ by $a^{9}$ by adding the exponents.

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Step 2.5.1.1

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

$\frac{a^{4 + 9} \frac{1}{b^{6}}}{b^{6}}$

Step 2.5.1.2

Add $4$ and $9$ .

$\frac{a^{13} \frac{1}{b^{6}}}{b^{6}}$

Step 2.5.2

Combine $a^{13}$ and $\frac{1}{b^{6}}$ .

$\frac{\frac{a^{13}}{b^{6}}}{b^{6}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{a^{13}}{b^{6}} \cdot \frac{1}{b^{6}}$

Step 4

Combine.

$\frac{a^{13} \cdot 1}{b^{6} b^{6}}$

Step 5

Multiply $b^{6}$ by $b^{6}$ by adding the exponents.

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Step 5.1

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

$\frac{a^{13} \cdot 1}{b^{6 + 6}}$

Step 5.2

Add $6$ and $6$ .

$\frac{a^{13} \cdot 1}{b^{12}}$

Step 6

Multiply $a^{13}$ by $1$ .

$\frac{a^{13}}{b^{12}}$