Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply the exponents in ${(2^{- 1})}^{2}$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $- 1$ by $2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Multiply $- 2$ by $3$ .

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

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

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

Step 2.9

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

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

Step 2.10

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

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Step 2.10.1

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

$\frac{1}{\frac{1}{4} a^{2} \frac{1}{a^{6}} b^{3} \frac{4}{a^{3 \cdot 2} b^{2}}}$

Step 2.10.2

Multiply $3$ by $2$ .

$\frac{1}{\frac{1}{4} a^{2} \frac{1}{a^{6}} b^{3} \frac{4}{a^{6} b^{2}}}$

Step 2.11

Combine exponents.

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Step 2.11.1

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

$\frac{1}{\frac{a^{2}}{4} \cdot \frac{1}{a^{6}} b^{3} \frac{4}{a^{6} b^{2}}}$

Step 2.11.2

Multiply $\frac{a^{2}}{4}$ by $\frac{1}{a^{6}}$ .

$\frac{1}{\frac{a^{2}}{4 a^{6}} b^{3} \frac{4}{a^{6} b^{2}}}$

Step 2.11.3

Combine $\frac{a^{2}}{4 a^{6}}$ and $b^{3}$ .

$\frac{1}{\frac{a^{2} b^{3}}{4 a^{6}} \cdot \frac{4}{a^{6} b^{2}}}$

Step 2.11.4

Multiply $\frac{a^{2} b^{3}}{4 a^{6}}$ by $\frac{4}{a^{6} b^{2}}$ .

$\frac{1}{\frac{a^{2} b^{3} \cdot 4}{4 a^{6} (a^{6} b^{2})}}$

Step 2.11.5

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

$\frac{1}{\frac{a^{2} b^{3} \cdot 4}{4 a^{6 + 6} b^{2}}}$

Step 2.11.6

Add $6$ and $6$ .

$\frac{1}{\frac{a^{2} b^{3} \cdot 4}{4 a^{12} b^{2}}}$

Step 2.12

Reduce the expression $\frac{a^{2} b^{3} \cdot 4}{4 a^{12} b^{2}}$ by cancelling the common factors.

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Step 2.12.1

Factor $a^{2}$ out of $a^{2} b^{3} \cdot 4$ .

$\frac{1}{\frac{a^{2} (b^{3} \cdot 4)}{4 a^{12} b^{2}}}$

Step 2.12.2

Factor $a^{2}$ out of $4 a^{12} b^{2}$ .

$\frac{1}{\frac{a^{2} (b^{3} \cdot 4)}{a^{2} (4 a^{10} b^{2})}}$

Step 2.12.3

Cancel the common factor.

$\frac{1}{\frac{a^{2} (b^{3} \cdot 4)}{a^{2} (4 a^{10} b^{2})}}$

Step 2.12.4

Rewrite the expression.

$\frac{1}{\frac{b^{3} \cdot 4}{4 a^{10} b^{2}}}$

Step 2.13

Cancel the common factor of $b^{3}$ and $b^{2}$ .

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Step 2.13.1

Factor $b^{2}$ out of $b^{3} \cdot 4$ .

$\frac{1}{\frac{b^{2} (b \cdot 4)}{4 a^{10} b^{2}}}$

Step 2.13.2

Cancel the common factors.

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Step 2.13.2.1

Factor $b^{2}$ out of $4 a^{10} b^{2}$ .

$\frac{1}{\frac{b^{2} (b \cdot 4)}{b^{2} (4 a^{10})}}$

Step 2.13.2.2

Cancel the common factor.

$\frac{1}{\frac{b^{2} (b \cdot 4)}{b^{2} (4 a^{10})}}$

Step 2.13.2.3

Rewrite the expression.

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

Step 2.14

Cancel the common factor of $4$ .

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Step 2.14.1

Cancel the common factor.

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

Step 2.14.2

Rewrite the expression.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{a^{10}}{b}$ by $1$ .

$\frac{a^{10}}{b}$