Basic Math Examples

$- a^{2 b^{- 2}} \div \frac{a^{- 2 b^{8}}}{a^{4 b^{- 2}}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$- a^{2 b^{- 2}} \frac{a^{4 b^{- 2}}}{a^{- 2 b^{8}}}$

Step 2

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

$- a^{2 \frac{1}{b^{2}}} \frac{a^{4 b^{- 2}}}{a^{- 2 b^{8}}}$

Step 3

Combine $2$ and $\frac{1}{b^{2}}$ .

$- a^{\frac{2}{b^{2}}} \frac{a^{4 b^{- 2}}}{a^{- 2 b^{8}}}$

Step 4

Cancel the common factor of $a^{4 b^{- 2}}$ and $a^{- 2 b^{8}}$ .

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Step 4.1

Factor $a^{- 2 b^{8}}$ out of $a^{4 b^{- 2}}$ .

$- a^{\frac{2}{b^{2}}} \frac{a^{- 2 b^{8}} a^{4 b^{- 2} + 2 b^{8}}}{a^{- 2 b^{8}}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$- a^{\frac{2}{b^{2}}} \frac{a^{- 2 b^{8}} a^{4 b^{- 2} + 2 b^{8}}}{a^{- 2 b^{8}} \cdot 1}$

Step 4.2.2

Cancel the common factor.

$- a^{\frac{2}{b^{2}}} \frac{a^{- 2 b^{8}} a^{4 b^{- 2} + 2 b^{8}}}{a^{- 2 b^{8}} \cdot 1}$

Step 4.2.3

Rewrite the expression.

$- a^{\frac{2}{b^{2}}} \frac{a^{4 b^{- 2} + 2 b^{8}}}{1}$

Step 4.2.4

Divide $a^{4 b^{- 2} + 2 b^{8}}$ by $1$ .

$- a^{\frac{2}{b^{2}}} a^{4 b^{- 2} + 2 b^{8}}$

Step 5

Multiply $a^{\frac{2}{b^{2}}}$ by $a^{4 b^{- 2} + 2 b^{8}}$ by adding the exponents.

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Step 5.1

Move $a^{4 b^{- 2} + 2 b^{8}}$ .

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

Step 5.2

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

$- a^{4 b^{- 2} + 2 b^{8} + \frac{2}{b^{2}}}$

Step 5.3

Simplify each term.

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Step 5.3.1

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

$- a^{4 \frac{1}{b^{2}} + 2 b^{8} + \frac{2}{b^{2}}}$

Step 5.3.2

Combine $4$ and $\frac{1}{b^{2}}$ .

$- a^{\frac{4}{b^{2}} + 2 b^{8} + \frac{2}{b^{2}}}$

Step 5.4

Combine the numerators over the common denominator.

$- a^{2 b^{8} + \frac{4 + 2}{b^{2}}}$

Step 5.5

Add $4$ and $2$ .

$- a^{2 b^{8} + \frac{6}{b^{2}}}$

Step 6

To write $2 b^{8}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

$- a^{\frac{2 b^{8} b^{2}}{b^{2}} + \frac{6}{b^{2}}}$

Step 7

Combine the numerators over the common denominator.

$- a^{\frac{2 b^{8} b^{2} + 6}{b^{2}}}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $2$ out of $2 b^{8} b^{2} + 6$ .

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Step 8.1.1

Factor $2$ out of $2 b^{8} b^{2}$ .

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

Step 8.1.2

Factor $2$ out of $6$ .

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

Step 8.1.3

Factor $2$ out of $2 (b^{8} b^{2}) + 2 \cdot 3$ .

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

Step 8.2

Multiply $b^{8}$ by $b^{2}$ by adding the exponents.

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Step 8.2.1

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

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

Step 8.2.2

Add $8$ and $2$ .

$- a^{\frac{2 (b^{10} + 3)}{b^{2}}}$