Simplify the Radical Expression (5^(1/2)*5^(-3/2))^(-1/4)

${(5^{\frac{1}{2}} \cdot 5^{- \frac{3}{2}})}^{- \frac{1}{4}}$

Step 1

Multiply $5^{\frac{1}{2}}$ by $5^{- \frac{3}{2}}$ by adding the exponents.

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

${(5^{\frac{1}{2} - \frac{3}{2}})}^{- \frac{1}{4}}$

Combine the numerators over the common denominator.

${(5^{\frac{1 - 3}{2}})}^{- \frac{1}{4}}$

Subtract $3$ from $1$ .

${(5^{\frac{- 2}{2}})}^{- \frac{1}{4}}$

Divide $- 2$ by $2$ .

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

Step 2

Multiply the exponents in ${(5^{- 1})}^{- \frac{1}{4}}$ .

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

$5^{- - \frac{1}{4}}$

Multiply $- - \frac{1}{4}$ .

Multiply $- 1$ by $- 1$ .

$5^{1 (\frac{1}{4})}$

Multiply $\frac{1}{4}$ by $1$ .

$5^{\frac{1}{4}}$