Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{m^{4}}{n^{6}} {(- m^{- 3} n^{3})}^{3}$

Step 4

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

$\frac{m^{4}}{n^{6}} {(- \frac{1}{m^{3}} n^{3})}^{3}$

Step 5

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

$\frac{m^{4}}{n^{6}} {(- \frac{n^{3}}{m^{3}})}^{3}$

Step 6

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

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Step 6.1

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

$\frac{m^{4}}{n^{6}} ({(- 1)}^{3} {(\frac{n^{3}}{m^{3}})}^{3})$

Step 6.2

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

$\frac{m^{4}}{n^{6}} ({(- 1)}^{3} \frac{{(n^{3})}^{3}}{{(m^{3})}^{3}})$

Step 7

Simplify the expression.

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Step 7.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{m^{4}}{n^{6}} \frac{{(n^{3})}^{3}}{{(m^{3})}^{3}}$

Step 7.2

Raise $- 1$ to the power of $3$ .

$- \frac{m^{4}}{n^{6}} \cdot \frac{{(n^{3})}^{3}}{{(m^{3})}^{3}}$

Step 7.3

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

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Step 7.3.1

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

$- \frac{m^{4}}{n^{6}} \cdot \frac{n^{3 \cdot 3}}{{(m^{3})}^{3}}$

Step 7.3.2

Multiply $3$ by $3$ .

$- \frac{m^{4}}{n^{6}} \cdot \frac{n^{9}}{{(m^{3})}^{3}}$

Step 7.4

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

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Step 7.4.1

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

$- \frac{m^{4}}{n^{6}} \cdot \frac{n^{9}}{m^{3 \cdot 3}}$

Step 7.4.2

Multiply $3$ by $3$ .

$- \frac{m^{4}}{n^{6}} \cdot \frac{n^{9}}{m^{9}}$

Step 8

Cancel the common factor of $m^{4}$ .

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Step 8.1

Move the leading negative in $- \frac{m^{4}}{n^{6}}$ into the numerator.

$\frac{- m^{4}}{n^{6}} \cdot \frac{n^{9}}{m^{9}}$

Step 8.2

Factor $m^{4}$ out of $- m^{4}$ .

$\frac{m^{4} \cdot - 1}{n^{6}} \cdot \frac{n^{9}}{m^{9}}$

Step 8.3

Factor $m^{4}$ out of $m^{9}$ .

$\frac{m^{4} \cdot - 1}{n^{6}} \cdot \frac{n^{9}}{m^{4} m^{5}}$

Step 8.4

Cancel the common factor.

$\frac{m^{4} \cdot - 1}{n^{6}} \cdot \frac{n^{9}}{m^{4} m^{5}}$

Step 8.5

Rewrite the expression.

$\frac{- 1}{n^{6}} \cdot \frac{n^{9}}{m^{5}}$

Step 9

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

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Step 9.1

Factor $n^{6}$ out of $n^{9}$ .

$\frac{- 1}{n^{6}} \cdot \frac{n^{6} n^{3}}{m^{5}}$

Step 9.2

Cancel the common factor.

$\frac{- 1}{n^{6}} \cdot \frac{n^{6} n^{3}}{m^{5}}$

Step 9.3

Rewrite the expression.

$- \frac{n^{3}}{m^{5}}$