Simplify the Radical Expression fifth root of x^7* cube root of x^2

$\sqrt[5]{x^{7}} \cdot \sqrt[3]{x^{2}}$

Step 1

Factor out $x^{5}$ .

$\sqrt[5]{x^{5} x^{2}} \cdot \sqrt[3]{x^{2}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[5]{x^{2}} \cdot \sqrt[3]{x^{2}}$

Step 3

Multiply $x \sqrt[5]{x^{2}} \sqrt[3]{x^{2}}$ .

Rewrite the expression using the least common index of $15$ .

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

$x (x^{\frac{2}{3}} \sqrt[5]{x^{2}})$

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{10}{15}}$ .

$x (x^{\frac{10}{15}} \sqrt[5]{x^{2}})$

Rewrite $x^{\frac{10}{15}}$ as $\sqrt[15]{x^{10}}$ .

$x (\sqrt[15]{x^{10}} \sqrt[5]{x^{2}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$x (\sqrt[15]{x^{10}} x^{\frac{2}{5}})$

Rewrite $x^{\frac{2}{5}}$ as $x^{\frac{6}{15}}$ .

$x (\sqrt[15]{x^{10}} x^{\frac{6}{15}})$

Rewrite $x^{\frac{6}{15}}$ as $\sqrt[15]{x^{6}}$ .

$x (\sqrt[15]{x^{10}} \sqrt[15]{x^{6}})$

Combine using the product rule for radicals.

$x \sqrt[15]{x^{10} x^{6}}$

Multiply $x^{10}$ by $x^{6}$ by adding the exponents.

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

$x \sqrt[15]{x^{10 + 6}}$

Add $10$ and $6$ .

$x \sqrt[15]{x^{16}}$

Step 4

Factor out $x^{15}$ .

$x \sqrt[15]{x^{15} x}$

Step 5

Pull terms out from under the radical.

$x (x \sqrt[15]{x})$

Step 6

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x \cdot x \sqrt[15]{x}$

Multiply $x$ by $x$ .

$x^{2} \sqrt[15]{x}$