Simplify the Radical Expression (x^(1/2)y^-3)^(1/2)

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

Step 1

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

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

Step 2

Combine $x^{\frac{1}{2}}$ and $\frac{1}{y^{3}}$ .

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

Step 3

Apply the product rule to $\frac{x^{\frac{1}{2}}}{y^{3}}$ .

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

Step 4

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

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

$\frac{x^{\frac{1}{2} \cdot \frac{1}{2}}}{{(y^{3})}^{\frac{1}{2}}}$

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{x^{\frac{1}{2 \cdot 2}}}{{(y^{3})}^{\frac{1}{2}}}$

Multiply $2$ by $2$ .

$\frac{x^{\frac{1}{4}}}{{(y^{3})}^{\frac{1}{2}}}$

Step 5

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

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

$\frac{x^{\frac{1}{4}}}{y^{3 (\frac{1}{2})}}$

Combine $3$ and $\frac{1}{2}$ .

$\frac{x^{\frac{1}{4}}}{y^{\frac{3}{2}}}$