Simplify the Radical Expression (x^16y^20z^8)^(1/4)

${(x^{16} y^{20} z^{8})}^{\frac{1}{4}}$

Step 1

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

Apply the product rule to $x^{16} y^{20} z^{8}$ .

${(x^{16} y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Apply the product rule to $x^{16} y^{20}$ .

${(x^{16})}^{\frac{1}{4}} {(y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Step 2

Multiply the exponents in ${(x^{16})}^{\frac{1}{4}}$ .

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

$x^{16 (\frac{1}{4})} {(y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

$x^{4 (4) \frac{1}{4}} {(y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Cancel the common factor.

$x^{4 \cdot 4 \frac{1}{4}} {(y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Rewrite the expression.

$x^{4} {(y^{20})}^{\frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Step 3

Multiply the exponents in ${(y^{20})}^{\frac{1}{4}}$ .

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

$x^{4} y^{20 (\frac{1}{4})} {(z^{8})}^{\frac{1}{4}}$

Cancel the common factor of $4$ .

Factor $4$ out of $20$ .

$x^{4} y^{4 (5) \frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Cancel the common factor.

$x^{4} y^{4 \cdot 5 \frac{1}{4}} {(z^{8})}^{\frac{1}{4}}$

Rewrite the expression.

$x^{4} y^{5} {(z^{8})}^{\frac{1}{4}}$

Step 4

Multiply the exponents in ${(z^{8})}^{\frac{1}{4}}$ .

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

$x^{4} y^{5} z^{8 (\frac{1}{4})}$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

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

Cancel the common factor.

$x^{4} y^{5} z^{4 \cdot 2 \frac{1}{4}}$

Rewrite the expression.

$x^{4} y^{5} z^{2}$