Simplify (32^2x^4y^12)^(1/5)

${(32^{2} x^{4} y^{12})}^{\frac{1}{5}}$

Step 1

Raise $32$ to the power of $2$ .

${(1024 x^{4} y^{12})}^{\frac{1}{5}}$

Step 2

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

Apply the product rule to $1024 x^{4} y^{12}$ .

${(1024 x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

Apply the product rule to $1024 x^{4}$ .

$1024^{\frac{1}{5}} {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

Step 3

Simplify the expression.

Rewrite $1024$ as $4^{5}$ .

${(4^{5})}^{\frac{1}{5}} {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

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

$4^{5 (\frac{1}{5})} {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

Step 4

Cancel the common factor of $5$ .

Cancel the common factor.

$4^{5 (\frac{1}{5})} {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

Rewrite the expression.

$4^{1} {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

Step 5

Simplify the expression.

Evaluate the exponent.

$4 {(x^{4})}^{\frac{1}{5}} {(y^{12})}^{\frac{1}{5}}$

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

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

$4 x^{4 (\frac{1}{5})} {(y^{12})}^{\frac{1}{5}}$

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

$4 x^{\frac{4}{5}} {(y^{12})}^{\frac{1}{5}}$

Multiply the exponents in ${(y^{12})}^{\frac{1}{5}}$ .

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

$4 x^{\frac{4}{5}} y^{12 (\frac{1}{5})}$

Combine $12$ and $\frac{1}{5}$ .

$4 x^{\frac{4}{5}} y^{\frac{12}{5}}$