Find the Product cube root of x^2* fourth root of x^3

$\sqrt[3]{x^{2}} \cdot \sqrt[4]{x^{3}}$

Step 1

Multiply $\sqrt[3]{x^{2}} \sqrt[4]{x^{3}}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$x^{\frac{2}{3}} \sqrt[4]{x^{3}}$

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$x^{\frac{8}{12}} \sqrt[4]{x^{3}}$

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$\sqrt[12]{x^{8}} \sqrt[4]{x^{3}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$\sqrt[12]{x^{8}} x^{\frac{3}{4}}$

Rewrite $x^{\frac{3}{4}}$ as $x^{\frac{9}{12}}$ .

$\sqrt[12]{x^{8}} x^{\frac{9}{12}}$

Rewrite $x^{\frac{9}{12}}$ as $\sqrt[12]{x^{9}}$ .

$\sqrt[12]{x^{8}} \sqrt[12]{x^{9}}$

Combine using the product rule for radicals.

$\sqrt[12]{x^{8} x^{9}}$

Multiply $x^{8}$ by $x^{9}$ by adding the exponents.

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

$\sqrt[12]{x^{8 + 9}}$

Add $8$ and $9$ .

$\sqrt[12]{x^{17}}$

Step 2

Factor out $x^{12}$ .

$\sqrt[12]{x^{12} x^{5}}$

Step 3

Pull terms out from under the radical.

$x \sqrt[12]{x^{5}}$