Simplify the Radical Expression (32x^4y^7)^(1/5)

${(32 x^{4} y^{7})}^{\frac{1}{5}}$

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

Apply the product rule to $32 x^{4} y^{7}$ .

${(32 x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

Apply the product rule to $32 x^{4}$ .

$32^{\frac{1}{5}} {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

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

${(2^{5})}^{\frac{1}{5}} {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

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

$2^{5 (\frac{1}{5})} {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$2^{5 (\frac{1}{5})} {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

Rewrite the expression.

$2^{1} {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

Evaluate the exponent.

$2 {(x^{4})}^{\frac{1}{5}} {(y^{7})}^{\frac{1}{5}}$

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

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

$2 x^{4 (\frac{1}{5})} {(y^{7})}^{\frac{1}{5}}$

Combine $4$ and $\frac{1}{5}$ .

$2 x^{\frac{4}{5}} {(y^{7})}^{\frac{1}{5}}$

Multiply the exponents in ${(y^{7})}^{\frac{1}{5}}$ .

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

$2 x^{\frac{4}{5}} y^{7 (\frac{1}{5})}$

Combine $7$ and $\frac{1}{5}$ .

$2 x^{\frac{4}{5}} y^{\frac{7}{5}}$