Simplify cube root of x* fourth root of x

$\sqrt[3]{x} \cdot \sqrt[4]{x}$

Step 1

Rewrite the expression using the least common index of $12$ .

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

$x^{\frac{1}{3}} \sqrt[4]{x}$

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{4}{12}}$ .

$x^{\frac{4}{12}} \sqrt[4]{x}$

Rewrite $x^{\frac{4}{12}}$ as $\sqrt[12]{x^{4}}$ .

$\sqrt[12]{x^{4}} \sqrt[4]{x}$

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

$\sqrt[12]{x^{4}} x^{\frac{1}{4}}$

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{3}{12}}$ .

$\sqrt[12]{x^{4}} x^{\frac{3}{12}}$

Rewrite $x^{\frac{3}{12}}$ as $\sqrt[12]{x^{3}}$ .

$\sqrt[12]{x^{4}} \sqrt[12]{x^{3}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[12]{x^{4} x^{3}}$

Step 3

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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

$\sqrt[12]{x^{4 + 3}}$

Add $4$ and $3$ .

$\sqrt[12]{x^{7}}$