Algebra Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $3 m^{- 2} n$ .

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

Step 2.2

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

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

Step 2.3

Combine $3$ and $\frac{1}{m^{2}}$ .

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

Step 2.4

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

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

Step 2.5

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

$\frac{n^{2}}{6 m \frac{27}{{(m^{2})}^{3}} n^{3}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{n^{2}}{6 m \frac{27}{m^{2 \cdot 3}} n^{3}}$

Step 2.6.2

Multiply $2$ by $3$ .

$\frac{n^{2}}{6 m \frac{27}{m^{6}} n^{3}}$

Step 2.7

Combine exponents.

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Step 2.7.1

Combine $6$ and $\frac{27}{m^{6}}$ .

$\frac{n^{2}}{m \frac{6 \cdot 27}{m^{6}} n^{3}}$

Step 2.7.2

Multiply $6$ by $27$ .

$\frac{n^{2}}{m \frac{162}{m^{6}} n^{3}}$

Step 2.7.3

Combine $m$ and $\frac{162}{m^{6}}$ .

$\frac{n^{2}}{\frac{m \cdot 162}{m^{6}} n^{3}}$

Step 2.7.4

Combine $\frac{m \cdot 162}{m^{6}}$ and $n^{3}$ .

$\frac{n^{2}}{\frac{m \cdot 162 n^{3}}{m^{6}}}$

Step 2.8

Reduce the expression $\frac{m \cdot 162 n^{3}}{m^{6}}$ by cancelling the common factors.

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Step 2.8.1

Factor $m$ out of $m \cdot 162 n^{3}$ .

$\frac{n^{2}}{\frac{m (162 n^{3})}{m^{6}}}$

Step 2.8.2

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

$\frac{n^{2}}{\frac{m (162 n^{3})}{m \cdot m^{5}}}$

Step 2.8.3

Cancel the common factor.

$\frac{n^{2}}{\frac{m (162 n^{3})}{m \cdot m^{5}}}$

Step 2.8.4

Rewrite the expression.

$\frac{n^{2}}{\frac{162 n^{3}}{m^{5}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$n^{2} \frac{m^{5}}{162 n^{3}}$

Step 4

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

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Step 4.1

Factor $n^{2}$ out of $162 n^{3}$ .

$n^{2} \frac{m^{5}}{n^{2} (162 n)}$

Step 4.2

Cancel the common factor.

$n^{2} \frac{m^{5}}{n^{2} (162 n)}$

Step 4.3

Rewrite the expression.

$\frac{m^{5}}{162 n}$