Simplify the Radical Expression (3x^(-1/3)y^(3/4))^2

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

Step 1

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

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

Step 2

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

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

Step 3

Combine $\frac{3}{x^{\frac{1}{3}}}$ and $y^{\frac{3}{4}}$ .

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3 y^{\frac{3}{4}}}{x^{\frac{1}{3}}}$ .

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

Apply the product rule to $3 y^{\frac{3}{4}}$ .

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

Step 5

Simplify the numerator.

Raise $3$ to the power of $2$ .

$\frac{9 {(y^{\frac{3}{4}})}^{2}}{{(x^{\frac{1}{3}})}^{2}}$

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

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

$\frac{9 y^{\frac{3}{4} \cdot 2}}{{(x^{\frac{1}{3}})}^{2}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{9 y^{\frac{3}{2 (2)} \cdot 2}}{{(x^{\frac{1}{3}})}^{2}}$

Cancel the common factor.

$\frac{9 y^{\frac{3}{2 \cdot 2} \cdot 2}}{{(x^{\frac{1}{3}})}^{2}}$

Rewrite the expression.

$\frac{9 y^{\frac{3}{2}}}{{(x^{\frac{1}{3}})}^{2}}$

Step 6

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

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

$\frac{9 y^{\frac{3}{2}}}{x^{\frac{1}{3} \cdot 2}}$

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

$\frac{9 y^{\frac{3}{2}}}{x^{\frac{2}{3}}}$