Simplify fourth root of x* fifth root of x

$\sqrt[4]{x} \cdot \sqrt[5]{x}$

Step 1

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

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

$x^{\frac{1}{4}} \sqrt[5]{x}$

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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\sqrt[20]{x^{5}} x^{\frac{1}{5}}$

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

$\sqrt[20]{x^{5}} x^{\frac{4}{20}}$

Rewrite $x^{\frac{4}{20}}$ as $\sqrt[20]{x^{4}}$ .

$\sqrt[20]{x^{5}} \sqrt[20]{x^{4}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[20]{x^{5} x^{4}}$

Step 3

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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

$\sqrt[20]{x^{5 + 4}}$

Add $5$ and $4$ .

$\sqrt[20]{x^{9}}$