Simplify ( fifth root of x^2)( fourth root of x)

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

Step 1

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 $x^{\frac{2}{5}}$ .

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

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

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

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

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

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

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

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

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

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

Add $8$ and $5$ .

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