Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Multiply the exponents in ${(a^{- \frac{m}{2}})}^{6}$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{m}{2}$ into the numerator.

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

Step 2.2.2.2

Factor $2$ out of $6$ .

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

Step 2.2.2.3

Cancel the common factor.

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

Step 2.2.2.4

Rewrite the expression.

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

Step 2.2.3

Multiply $3$ by $- 1$ .

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

Step 2.3

Multiply the exponents in ${(b^{- \frac{n}{3}})}^{6}$ .

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $3$ .

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Step 2.3.2.1

Move the leading negative in $- \frac{n}{3}$ into the numerator.

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

Step 2.3.2.2

Factor $3$ out of $6$ .

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

Step 2.3.2.3

Cancel the common factor.

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

Step 2.3.2.4

Rewrite the expression.

$\frac{1}{a^{- 3 m} b^{- n \cdot 2}}$

Step 2.3.3

Multiply $2$ by $- 1$ .

$\frac{1}{a^{- 3 m} b^{- 2 n}}$