Basic Math Examples

${(6 m^{2} n^{- 2})}^{- 3} {(6 m^{- 1} n)}^{- 2}$

Step 1

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

${(6 m^{2} \frac{1}{n^{2}})}^{- 3} {(6 m^{- 1} n)}^{- 2}$

Step 2

Multiply $6 m^{2} \frac{1}{n^{2}}$ .

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Step 2.1

Combine $6$ and $\frac{1}{n^{2}}$ .

${(m^{2} \frac{6}{n^{2}})}^{- 3} {(6 m^{- 1} n)}^{- 2}$

Step 2.2

Combine $m^{2}$ and $\frac{6}{n^{2}}$ .

${(\frac{m^{2} \cdot 6}{n^{2}})}^{- 3} {(6 m^{- 1} n)}^{- 2}$

Step 3

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

${(\frac{6 \cdot m^{2}}{n^{2}})}^{- 3} {(6 m^{- 1} n)}^{- 2}$

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{n^{2}}{6 m^{2}})}^{3} {(6 m^{- 1} n)}^{- 2}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{n^{2}}{6 m^{2}}$ .

$\frac{{(n^{2})}^{3}}{{(6 m^{2})}^{3}} {(6 m^{- 1} n)}^{- 2}$

Step 5.2

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

$\frac{{(n^{2})}^{3}}{6^{3} {(m^{2})}^{3}} {(6 m^{- 1} n)}^{- 2}$

Step 6

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

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Step 6.1

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

$\frac{n^{2 \cdot 3}}{6^{3} {(m^{2})}^{3}} {(6 m^{- 1} n)}^{- 2}$

Step 6.2

Multiply $2$ by $3$ .

$\frac{n^{6}}{6^{3} {(m^{2})}^{3}} {(6 m^{- 1} n)}^{- 2}$

Step 7

Simplify the denominator.

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Step 7.1

Raise $6$ to the power of $3$ .

$\frac{n^{6}}{216 {(m^{2})}^{3}} {(6 m^{- 1} n)}^{- 2}$

Step 7.2

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

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Step 7.2.1

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

$\frac{n^{6}}{216 m^{2 \cdot 3}} {(6 m^{- 1} n)}^{- 2}$

Step 7.2.2

Multiply $2$ by $3$ .

$\frac{n^{6}}{216 m^{6}} {(6 m^{- 1} n)}^{- 2}$

Step 8

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

$\frac{n^{6}}{216 m^{6}} {(6 \frac{1}{m} n)}^{- 2}$

Step 9

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

$\frac{n^{6}}{216 m^{6}} {(\frac{6}{m} n)}^{- 2}$

Step 10

Combine $\frac{6}{m}$ and $n$ .

$\frac{n^{6}}{216 m^{6}} {(\frac{6 n}{m})}^{- 2}$

Step 11

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{n^{6}}{216 m^{6}} {(\frac{m}{6 n})}^{2}$

Step 12

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

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Step 12.1

Apply the product rule to $\frac{m}{6 n}$ .

$\frac{n^{6}}{216 m^{6}} \cdot \frac{m^{2}}{{(6 n)}^{2}}$

Step 12.2

Apply the product rule to $6 n$ .

$\frac{n^{6}}{216 m^{6}} \cdot \frac{m^{2}}{6^{2} n^{2}}$

Step 13

Simplify terms.

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Step 13.1

Combine.

$\frac{n^{6} m^{2}}{216 m^{6} (6^{2} n^{2})}$

Step 13.2

Cancel the common factor of $n^{6}$ and $n^{2}$ .

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Step 13.2.1

Factor $n^{2}$ out of $n^{6} m^{2}$ .

$\frac{n^{2} (n^{4} m^{2})}{216 m^{6} (6^{2} n^{2})}$

Step 13.2.2

Cancel the common factors.

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Step 13.2.2.1

Factor $n^{2}$ out of $216 m^{6} (6^{2} n^{2})$ .

$\frac{n^{2} (n^{4} m^{2})}{n^{2} (216 m^{6} \cdot (6^{2}))}$

Step 13.2.2.2

Cancel the common factor.

$\frac{n^{2} (n^{4} m^{2})}{n^{2} (216 m^{6} \cdot (6^{2}))}$

Step 13.2.2.3

Rewrite the expression.

$\frac{n^{4} m^{2}}{216 m^{6} \cdot (6^{2})}$

Step 13.3

Cancel the common factor of $m^{2}$ and $m^{6}$ .

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Step 13.3.1

Factor $m^{2}$ out of $n^{4} m^{2}$ .

$\frac{m^{2} n^{4}}{216 m^{6} \cdot (6^{2})}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $m^{2}$ out of $216 m^{6} \cdot (6^{2})$ .

$\frac{m^{2} n^{4}}{m^{2} (216 m^{4} \cdot 6^{2})}$

Step 13.3.2.2

Cancel the common factor.

$\frac{m^{2} n^{4}}{m^{2} (216 m^{4} \cdot 6^{2})}$

Step 13.3.2.3

Rewrite the expression.

$\frac{n^{4}}{216 m^{4} \cdot 6^{2}}$

Step 14

Simplify the denominator.

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Step 14.1

Rewrite $216$ as $6^{3}$ .

$\frac{n^{4}}{6^{2} \cdot 6^{3} m^{4}}$

Step 14.2

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

$\frac{n^{4}}{6^{2 + 3} m^{4}}$

Step 14.3

Add $2$ and $3$ .

$\frac{n^{4}}{6^{5} m^{4}}$

Step 15

Raise $6$ to the power of $5$ .

$\frac{n^{4}}{7776 m^{4}}$