Find the Product cube root of x* square root of 2x

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

Step 1

Multiply $\sqrt[3]{x} \sqrt{2 x}$ .

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

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

$x^{\frac{1}{3}} \sqrt{2 x}$

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

$x^{\frac{2}{6}} \sqrt{2 x}$

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

$\sqrt[6]{x^{2}} \sqrt{2 x}$

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

$\sqrt[6]{x^{2}} {(2 x)}^{\frac{1}{2}}$

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

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

Rewrite ${(2 x)}^{\frac{3}{6}}$ as $\sqrt[6]{{(2 x)}^{3}}$ .

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

Combine using the product rule for radicals.

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

Apply the product rule to $2 x$ .

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

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[6]{x^{3 + 2} \cdot 2^{3}}$

Add $3$ and $2$ .

$\sqrt[6]{x^{5} \cdot 2^{3}}$

Raise $2$ to the power of $3$ .

$\sqrt[6]{x^{5} \cdot 8}$

Step 2

Reorder factors in $\sqrt[6]{x^{5} \cdot 8}$ .

$\sqrt[6]{8 x^{5}}$