Simplify the Radical Expression (16 square root of x^3)^(1/4)

${(16 \sqrt{x^{3}})}^{\frac{1}{4}}$

Factor out $x^{2}$ .

${(16 \sqrt{x^{2} x})}^{\frac{1}{4}}$

Pull terms out from under the radical.

${(16 (| x | \sqrt{x}))}^{\frac{1}{4}}$

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

Apply the product rule to $16 | x | \sqrt{x}$ .

${(16 | x |)}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

Apply the product rule to $16 | x |$ .

$16^{\frac{1}{4}} {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

Rewrite $16$ as $2^{4}$ .

${(2^{4})}^{\frac{1}{4}} {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

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

$2^{4 (\frac{1}{4})} {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$2^{4 (\frac{1}{4})} {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

Rewrite the expression.

$2^{1} {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$

Evaluate the exponent.

$2 {| x |}^{\frac{1}{4}} {\sqrt{x}}^{\frac{1}{4}}$