Algebra Examples

${(2^{- 1} m^{2} n)}^{- 3} \cdot (\frac{m^{0} n^{- 4}}{4 m^{- 1} n})$

Step 1

Simplify the numerator.

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Step 1.1

Anything raised to $0$ is $1$ .

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{1 n^{- 4}}{4 m^{- 1} n}$

Step 1.2

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

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{1 \frac{1}{n^{4}}}{4 m^{- 1} n}$

Step 1.3

Multiply $\frac{1}{n^{4}}$ by $1$ .

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{4 m^{- 1} n}$

Step 2

Simplify the denominator.

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Step 2.1

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

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{4 \frac{1}{m} n}$

Step 2.2

Combine exponents.

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Step 2.2.1

Combine $4$ and $\frac{1}{m}$ .

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4}{m} n}$

Step 2.2.2

Combine $\frac{4}{m}$ and $n$ .

${(2^{- 1} m^{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 3

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

${(\frac{1}{2} m^{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 4

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

${(\frac{m^{2}}{2} n)}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 5

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

${(\frac{m^{2} n}{2})}^{- 3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 6

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

${(\frac{2}{m^{2} n})}^{3} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 7

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

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Step 7.1

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

$\frac{2^{3}}{{(m^{2} n)}^{3}} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 7.2

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

$\frac{2^{3}}{{(m^{2})}^{3} n^{3}} \cdot \frac{\frac{1}{n^{4}}}{\frac{4 n}{m}}$

Step 8

Combine fractions.

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Step 8.1

Combine.

$\frac{2^{3} \frac{1}{n^{4}}}{{(m^{2})}^{3} n^{3} \frac{4 n}{m}}$

Step 8.2

Combine $2^{3}$ and $\frac{1}{n^{4}}$ .

$\frac{\frac{2^{3}}{n^{4}}}{{(m^{2})}^{3} n^{3} \frac{4 n}{m}}$

Step 9

Simplify the denominator.

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Step 9.1

Combine ${(m^{2})}^{3}$ and $\frac{4 n}{m}$ .

$\frac{\frac{2^{3}}{n^{4}}}{n^{3} \frac{{(m^{2})}^{3} (4 n)}{m}}$

Step 9.2

Combine $n^{3}$ and $\frac{{(m^{2})}^{3} (4 n)}{m}$ .

$\frac{\frac{2^{3}}{n^{4}}}{\frac{n^{3} ({(m^{2})}^{3} (4 n))}{m}}$

Step 10

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

$\frac{\frac{8}{n^{4}}}{\frac{n^{3} ({(m^{2})}^{3} (4 n))}{m}}$

Step 11

Multiply $n^{3}$ by $n$ by adding the exponents.

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Step 11.1

Move $n$ .

$\frac{\frac{8}{n^{4}}}{\frac{n \cdot n^{3} ({(m^{2})}^{3} \cdot 4)}{m}}$

Step 11.2

Multiply $n$ by $n^{3}$ .

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Step 11.2.1

Raise $n$ to the power of $1$ .

$\frac{\frac{8}{n^{4}}}{\frac{n^{1} n^{3} ({(m^{2})}^{3} \cdot 4)}{m}}$

Step 11.2.2

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

$\frac{\frac{8}{n^{4}}}{\frac{n^{1 + 3} ({(m^{2})}^{3} \cdot 4)}{m}}$

Step 11.3

Add $1$ and $3$ .

$\frac{\frac{8}{n^{4}}}{\frac{n^{4} ({(m^{2})}^{3} \cdot 4)}{m}}$

Step 12

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

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Step 12.1

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

$\frac{\frac{8}{n^{4}}}{\frac{n^{4} m^{2 \cdot 3} \cdot 4}{m}}$

Step 12.2

Multiply $2$ by $3$ .

$\frac{\frac{8}{n^{4}}}{\frac{n^{4} m^{6} \cdot 4}{m}}$

Step 13

Reduce the expression by cancelling the common factors.

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Step 13.1

Reduce the expression $\frac{n^{4} m^{6} \cdot 4}{m}$ by cancelling the common factors.

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Step 13.1.1

Factor $m$ out of $n^{4} m^{6} \cdot 4$ .

$\frac{\frac{8}{n^{4}}}{\frac{m (n^{4} m^{5} \cdot 4)}{m}}$

Step 13.1.2

Raise $m$ to the power of $1$ .

$\frac{\frac{8}{n^{4}}}{\frac{m (n^{4} m^{5} \cdot 4)}{m^{1}}}$

Step 13.1.3

Factor $m$ out of $m^{1}$ .

$\frac{\frac{8}{n^{4}}}{\frac{m (n^{4} m^{5} \cdot 4)}{m \cdot 1}}$

Step 13.1.4

Cancel the common factor.

$\frac{\frac{8}{n^{4}}}{\frac{m (n^{4} m^{5} \cdot 4)}{m \cdot 1}}$

Step 13.1.5

Rewrite the expression.

$\frac{\frac{8}{n^{4}}}{\frac{n^{4} m^{5} \cdot 4}{1}}$

Step 13.2

Divide $n^{4} m^{5} \cdot 4$ by $1$ .

$\frac{\frac{8}{n^{4}}}{n^{4} m^{5} \cdot 4}$

Step 14

Multiply the numerator by the reciprocal of the denominator.

$\frac{8}{n^{4}} \cdot \frac{1}{n^{4} m^{5} \cdot 4}$

Step 15

Combine.

$\frac{8 \cdot 1}{n^{4} (n^{4} m^{5} \cdot 4)}$

Step 16

Multiply $n^{4}$ by $n^{4}$ by adding the exponents.

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Step 16.1

Move $n^{4}$ .

$\frac{8 \cdot 1}{n^{4} n^{4} (m^{5} \cdot 4)}$

Step 16.2

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

$\frac{8 \cdot 1}{n^{4 + 4} (m^{5} \cdot 4)}$

Step 16.3

Add $4$ and $4$ .

$\frac{8 \cdot 1}{n^{8} (m^{5} \cdot 4)}$

Step 17

Multiply $8$ by $1$ .

$\frac{8}{n^{8} (m^{5} \cdot 4)}$

Step 18

Cancel the common factor of $8$ and $4$ .

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Step 18.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2}{n^{8} (m^{5} \cdot 4)}$

Step 18.2

Cancel the common factors.

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Step 18.2.1

Factor $4$ out of $n^{8} (m^{5} \cdot 4)$ .

$\frac{4 \cdot 2}{4 (n^{8} (m^{5}))}$

Step 18.2.2

Cancel the common factor.

$\frac{4 \cdot 2}{4 (n^{8} (m^{5}))}$

Step 18.2.3

Rewrite the expression.

$\frac{2}{n^{8} (m^{5})}$

$\frac{2}{n^{8} m^{5}}$