Simplify fourth root of x^5 fifth root of x^2

$\sqrt[4]{x^{5}} \sqrt[5]{x^{2}}$

Step 1

Factor out $x^{4}$ .

$\sqrt[4]{x^{4} x} \sqrt[5]{x^{2}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[4]{x} \sqrt[5]{x^{2}}$

Step 3

Multiply $x \sqrt[4]{x} \sqrt[5]{x^{2}}$ .

Rewrite the expression using the least common index of $20$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$x (x^{\frac{2}{5}} \sqrt[4]{x})$

Rewrite $x^{\frac{2}{5}}$ as $x^{\frac{8}{20}}$ .

$x (x^{\frac{8}{20}} \sqrt[4]{x})$

Rewrite $x^{\frac{8}{20}}$ as $\sqrt[20]{x^{8}}$ .

$x (\sqrt[20]{x^{8}} \sqrt[4]{x})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$x (\sqrt[20]{x^{8}} x^{\frac{1}{4}})$

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{5}{20}}$ .

$x (\sqrt[20]{x^{8}} x^{\frac{5}{20}})$

Rewrite $x^{\frac{5}{20}}$ as $\sqrt[20]{x^{5}}$ .

$x (\sqrt[20]{x^{8}} \sqrt[20]{x^{5}})$

Combine using the product rule for radicals.

$x \sqrt[20]{x^{8} x^{5}}$

Multiply $x^{8}$ by $x^{5}$ by adding the exponents.

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

$x \sqrt[20]{x^{8 + 5}}$

Add $8$ and $5$ .

$x \sqrt[20]{x^{13}}$