Simplify square root of x fifth root of x

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

Step 1

Rewrite the expression using the least common index of $10$ .

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

$x^{\frac{1}{2}} \sqrt[5]{x}$

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{5}{10}}$ .

$x^{\frac{5}{10}} \sqrt[5]{x}$

Rewrite $x^{\frac{5}{10}}$ as $\sqrt[10]{x^{5}}$ .

$\sqrt[10]{x^{5}} \sqrt[5]{x}$

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

$\sqrt[10]{x^{5}} x^{\frac{1}{5}}$

Rewrite $x^{\frac{1}{5}}$ as $x^{\frac{2}{10}}$ .

$\sqrt[10]{x^{5}} x^{\frac{2}{10}}$

Rewrite $x^{\frac{2}{10}}$ as $\sqrt[10]{x^{2}}$ .

$\sqrt[10]{x^{5}} \sqrt[10]{x^{2}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[10]{x^{5} x^{2}}$

Step 3

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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

$\sqrt[10]{x^{5 + 2}}$

Add $5$ and $2$ .

$\sqrt[10]{x^{7}}$