Divide ((32^(4/5))/(32^(3/5)))^-1

${(\frac{32^{\frac{4}{5}}}{32^{\frac{3}{5}}})}^{- 1}$

Step 1

Move $32^{\frac{3}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(32^{\frac{4}{5}} \cdot 32^{- \frac{3}{5}})}^{- 1}$

Step 2

Multiply $32^{\frac{4}{5}}$ by $32^{- \frac{3}{5}}$ by adding the exponents.

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

${(32^{\frac{4}{5} - \frac{3}{5}})}^{- 1}$

Combine the numerators over the common denominator.

${(32^{\frac{4 - 3}{5}})}^{- 1}$

Subtract $3$ from $4$ .

${(32^{\frac{1}{5}})}^{- 1}$

Step 3

Multiply the exponents in ${(32^{\frac{1}{5}})}^{- 1}$ .

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

$32^{\frac{1}{5} \cdot - 1}$

Combine $\frac{1}{5}$ and $- 1$ .

$32^{\frac{- 1}{5}}$

Move the negative in front of the fraction.

$32^{- \frac{1}{5}}$

Step 4

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

$\frac{1}{32^{\frac{1}{5}}}$

Step 5

Simplify the denominator.

Rewrite $32$ as $2^{5}$ .

$\frac{1}{{(2^{5})}^{\frac{1}{5}}}$

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

$\frac{1}{2^{5 (\frac{1}{5})}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{1}{2^{5 (\frac{1}{5})}}$

Rewrite the expression.

$\frac{1}{2^{1}}$

Evaluate the exponent.

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$