Precalculus Examples

$\frac{{(3^{2})}^{\frac{- 1}{2}} {(9^{4})}^{- 1}}{27^{- 3}}$

Step 1

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

$\frac{{(9^{4})}^{- 1}}{27^{- 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 2

Move ${(9^{4})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{27^{- 3} {(3^{2})}^{- \frac{- 1}{2}} \cdot 9^{4}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{{(3^{2})}^{4} \cdot 27^{- 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.2

Multiply the exponents in ${(3^{2})}^{4}$ .

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Step 3.2.1

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

$\frac{1}{3^{2 \cdot 4} \cdot 27^{- 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.2.2

Multiply $2$ by $4$ .

$\frac{1}{3^{8} \cdot 27^{- 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.3

Rewrite $27$ as $3^{3}$ .

$\frac{1}{3^{8} \cdot {(3^{3})}^{- 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{1}{3^{8} \cdot 3^{3 \cdot - 3} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.4.2

Multiply $3$ by $- 3$ .

$\frac{1}{3^{8} \cdot 3^{- 9} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.5

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

$\frac{1}{3^{8 - 9} {(3^{2})}^{- \frac{- 1}{2}}}$

Step 3.6

Subtract $9$ from $8$ .

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

Step 4

Move $3^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Multiply the exponents in ${(3^{2})}^{- \frac{- 1}{2}}$ .

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Step 5.1.1

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

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

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Move the leading negative in $- \frac{- 1}{2}$ into the numerator.

$\frac{3}{3^{2 (\frac{1}{2})}}$

Step 5.1.2.2

Cancel the common factor.

$\frac{3}{3^{2 (\frac{1}{2})}}$

Step 5.1.2.3

Rewrite the expression.

$\frac{3}{3^{1}}$

Step 5.2

Evaluate the exponent.

$\frac{3}{3}$

Step 6

Divide $3$ by $3$ .

$1$