Simplify the Radical Expression cube root of 64x* fourth root of 16x^2

$\sqrt[3]{64 x} \cdot \sqrt[4]{16 x^{2}}$

Step 1

Rewrite $64$ as $4^{3}$ .

$\sqrt[3]{4^{3} x} \cdot \sqrt[4]{16 x^{2}}$

Step 2

Pull terms out from under the radical.

$4 \sqrt[3]{x} \cdot \sqrt[4]{16 x^{2}}$

Step 3

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

$4 \sqrt[3]{x} \cdot \sqrt[4]{2^{4} x^{2}}$

Step 4

Pull terms out from under the radical.

$4 \sqrt[3]{x} \cdot (| 2 | \sqrt[4]{x^{2}})$

Step 5

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$4 \sqrt[3]{x} \cdot (2 \sqrt[4]{x^{2}})$

Step 6

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$4 \sqrt[3]{x} \cdot (2 \sqrt{\sqrt{x^{2}}})$

Step 7

Pull terms out from under the radical.

$4 \sqrt[3]{x} \cdot (2 \sqrt{| x |})$

Step 8

Multiply $4 \sqrt[3]{x} (2 \sqrt{| x |})$ .

Multiply $2$ by $4$ .

$8 \sqrt[3]{x} \sqrt{| x |}$

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{| x |}$ as ${| x |}^{\frac{1}{2}}$ .

$8 ({| x |}^{\frac{1}{2}} \sqrt[3]{x})$

Rewrite ${| x |}^{\frac{1}{2}}$ as ${| x |}^{\frac{3}{6}}$ .

$8 ({| x |}^{\frac{3}{6}} \sqrt[3]{x})$

Rewrite ${| x |}^{\frac{3}{6}}$ as $\sqrt[6]{{| x |}^{3}}$ .

$8 (\sqrt[6]{{| x |}^{3}} \sqrt[3]{x})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$8 (\sqrt[6]{{| x |}^{3}} x^{\frac{1}{3}})$

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$8 (\sqrt[6]{{| x |}^{3}} x^{\frac{2}{6}})$

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$8 (\sqrt[6]{{| x |}^{3}} \sqrt[6]{x^{2}})$

Combine using the product rule for radicals.

$8 \sqrt[6]{{| x |}^{3} x^{2}}$

Step 9

Reorder factors in $8 \sqrt[6]{{| x |}^{3} x^{2}}$ .

$8 \sqrt[6]{x^{2} {| x |}^{3}}$