Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

Multiply $3$ by $2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $4 m^{- 4} n$ .

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

Step 2.2

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

$\frac{\frac{4}{m^{6}} n^{2}}{{(4 \frac{1}{m^{4}})}^{2} n^{2}}$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Raise $4$ to the power of $2$ .

$\frac{\frac{4}{m^{6}} n^{2}}{\frac{16}{{(m^{4})}^{2}} n^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{\frac{4}{m^{6}} n^{2}}{\frac{16}{m^{4 \cdot 2}} n^{2}}$

Step 2.6.2

Multiply $4$ by $2$ .

$\frac{\frac{4}{m^{6}} n^{2}}{\frac{16}{m^{8}} n^{2}}$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{4}{m^{6}}$ and $n^{2}$ .

$\frac{\frac{4 n^{2}}{m^{6}}}{\frac{16}{m^{8}} n^{2}}$

Step 3.2

Combine $\frac{16}{m^{8}}$ and $n^{2}$ .

$\frac{\frac{4 n^{2}}{m^{6}}}{\frac{16 n^{2}}{m^{8}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 n^{2}}{m^{6}} \cdot \frac{m^{8}}{16 n^{2}}$

Step 5

Combine.

$\frac{4 n^{2} m^{8}}{m^{6} (16 n^{2})}$

Step 6

Cancel the common factor of $4$ and $16$ .

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Step 6.1

Factor $4$ out of $4 n^{2} m^{8}$ .

$\frac{4 (n^{2} m^{8})}{m^{6} (16 n^{2})}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $4$ out of $m^{6} (16 n^{2})$ .

$\frac{4 (n^{2} m^{8})}{4 (m^{6} (4 n^{2}))}$

Step 6.2.2

Cancel the common factor.

$\frac{4 (n^{2} m^{8})}{4 (m^{6} (4 n^{2}))}$

Step 6.2.3

Rewrite the expression.

$\frac{n^{2} m^{8}}{m^{6} (4 n^{2})}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{n^{2} m^{8}}{m^{6} (4 n^{2})}$

Step 7.2

Rewrite the expression.

$\frac{m^{8}}{m^{6} \cdot (4)}$

Step 8

Cancel the common factor of $m^{8}$ and $m^{6}$ .

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Step 8.1

Factor $m^{6}$ out of $m^{8}$ .

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

$\frac{m^{2}}{4}$