Basic Math Examples

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

Step 1

Move ${(3^{- 1} a^{4} b^{- 3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $6 a^{2} b^{- 1} c^{- 2}$ .

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

Step 2.2

Apply the product rule to $3^{- 1} a^{4} b^{- 3}$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

Combine $6$ and $\frac{1}{b}$ .

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

Step 2.4.2

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

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

Step 2.5

Move $6$ to the left of $a^{2}$ .

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

Step 2.6

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

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Step 2.6.1

Apply the product rule to $\frac{6 a^{2}}{b}$ .

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

Step 2.6.2

Apply the product rule to $6 a^{2}$ .

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Raise $6$ to the power of $2$ .

$\frac{1}{\frac{36 {(a^{2})}^{2}}{b^{2}} {(c^{- 2})}^{2} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.7.2

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

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Step 2.7.2.1

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

$\frac{1}{\frac{36 a^{2 \cdot 2}}{b^{2}} {(c^{- 2})}^{2} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.7.2.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{36 a^{4}}{b^{2}} {(c^{- 2})}^{2} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} c^{- 2 \cdot 2} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.8.2

Multiply $- 2$ by $2$ .

$\frac{1}{\frac{36 a^{4}}{b^{2}} c^{- 4} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.9

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} {(3^{- 1} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.10

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} {(\frac{1}{3} a^{4})}^{2} {(b^{- 3})}^{2}}$

Step 2.11

Combine $\frac{1}{3}$ and $a^{4}$ .

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} {(\frac{a^{4}}{3})}^{2} {(b^{- 3})}^{2}}$

Step 2.12

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{{(a^{4})}^{2}}{3^{2}} {(b^{- 3})}^{2}}$

Step 2.13

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

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Step 2.13.1

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{4 \cdot 2}}{3^{2}} {(b^{- 3})}^{2}}$

Step 2.13.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{8}}{3^{2}} {(b^{- 3})}^{2}}$

Step 2.14

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{8}}{9} {(b^{- 3})}^{2}}$

Step 2.15

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

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Step 2.15.1

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{8}}{9} b^{- 3 \cdot 2}}$

Step 2.15.2

Multiply $- 3$ by $2$ .

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{8}}{9} b^{- 6}}$

Step 2.16

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

$\frac{1}{\frac{36 a^{4}}{b^{2}} \cdot \frac{1}{c^{4}} \frac{a^{8}}{9} \frac{1}{b^{6}}}$

Step 2.17

Combine exponents.

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Step 2.17.1

Multiply $\frac{36 a^{4}}{b^{2}}$ by $\frac{1}{c^{4}}$ .

$\frac{1}{\frac{36 a^{4}}{b^{2} c^{4}} \cdot \frac{a^{8}}{9} \frac{1}{b^{6}}}$

Step 2.17.2

Multiply $\frac{36 a^{4}}{b^{2} c^{4}}$ by $\frac{a^{8}}{9}$ .

$\frac{1}{\frac{36 a^{4} a^{8}}{b^{2} c^{4} \cdot 9} \cdot \frac{1}{b^{6}}}$

Step 2.17.3

Multiply $a^{4}$ by $a^{8}$ by adding the exponents.

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Step 2.17.3.1

Move $a^{8}$ .

$\frac{1}{\frac{36 (a^{8} a^{4})}{b^{2} c^{4} \cdot 9} \cdot \frac{1}{b^{6}}}$

Step 2.17.3.2

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

$\frac{1}{\frac{36 a^{8 + 4}}{b^{2} c^{4} \cdot 9} \cdot \frac{1}{b^{6}}}$

Step 2.17.3.3

Add $8$ and $4$ .

$\frac{1}{\frac{36 a^{12}}{b^{2} c^{4} \cdot 9} \cdot \frac{1}{b^{6}}}$

Step 2.17.4

Multiply $\frac{36 a^{12}}{b^{2} c^{4} \cdot 9}$ by $\frac{1}{b^{6}}$ .

$\frac{1}{\frac{36 a^{12}}{b^{2} c^{4} \cdot 9 b^{6}}}$

Step 2.17.5

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

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Step 2.17.5.1

Move $b^{6}$ .

$\frac{1}{\frac{36 a^{12}}{b^{6} b^{2} c^{4} \cdot 9}}$

Step 2.17.5.2

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

$\frac{1}{\frac{36 a^{12}}{b^{6 + 2} c^{4} \cdot 9}}$

Step 2.17.5.3

Add $6$ and $2$ .

$\frac{1}{\frac{36 a^{12}}{b^{8} c^{4} \cdot 9}}$

Step 2.18

Reduce the expression $\frac{36 a^{12}}{b^{8} c^{4} \cdot 9}$ by cancelling the common factors.

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Step 2.18.1

Factor $9$ out of $36 a^{12}$ .

$\frac{1}{\frac{9 (4 a^{12})}{b^{8} c^{4} \cdot 9}}$

Step 2.18.2

Factor $9$ out of $b^{8} c^{4} \cdot 9$ .

$\frac{1}{\frac{9 (4 a^{12})}{9 \cdot (b^{8} c^{4})}}$

Step 2.18.3

Cancel the common factor.

$\frac{1}{\frac{9 (4 a^{12})}{9 \cdot (b^{8} c^{4})}}$

Step 2.18.4

Rewrite the expression.

$\frac{1}{\frac{4 a^{12}}{b^{8} c^{4}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{b^{8} c^{4}}{4 a^{12}}$

Step 4

Multiply $\frac{b^{8} c^{4}}{4 a^{12}}$ by $1$ .

$\frac{b^{8} c^{4}}{4 a^{12}}$