Simplify cube root of 125x^8* fourth root of 16x

$\sqrt[3]{125 x^{8}} \cdot \sqrt[4]{16 x}$

Rewrite $125 x^{8}$ as ${(5 x^{2})}^{3} x^{2}$ .

Rewrite $125$ as $5^{3}$ .

$\sqrt[3]{5^{3} x^{8}} \cdot \sqrt[4]{16 x}$

Factor out $x^{6}$ .

$\sqrt[3]{5^{3} (x^{6} x^{2})} \cdot \sqrt[4]{16 x}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\sqrt[3]{5^{3} ({(x^{2})}^{3} x^{2})} \cdot \sqrt[4]{16 x}$

Rewrite $5^{3} {(x^{2})}^{3}$ as ${(5 x^{2})}^{3}$ .

$\sqrt[3]{{(5 x^{2})}^{3} x^{2}} \cdot \sqrt[4]{16 x}$

Pull terms out from under the radical.

$5 x^{2} \sqrt[3]{x^{2}} \cdot \sqrt[4]{16 x}$

Rewrite $16$ as $2^{4}$ .

$5 x^{2} \sqrt[3]{x^{2}} \cdot \sqrt[4]{2^{4} x}$

Pull terms out from under the radical.

$5 x^{2} \sqrt[3]{x^{2}} \cdot (2 \sqrt[4]{x})$

Multiply $5 x^{2} \sqrt[3]{x^{2}} (2 \sqrt[4]{x})$ .

Multiply $2$ by $5$ .

$10 x^{2} \sqrt[3]{x^{2}} \sqrt[4]{x}$

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

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

$10 x^{2} (x^{\frac{1}{4}} \sqrt[3]{x^{2}})$

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

$10 x^{2} (x^{\frac{3}{12}} \sqrt[3]{x^{2}})$

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

$10 x^{2} (\sqrt[12]{x^{3}} \sqrt[3]{x^{2}})$

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

$10 x^{2} (\sqrt[12]{x^{3}} x^{\frac{2}{3}})$

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

$10 x^{2} (\sqrt[12]{x^{3}} x^{\frac{8}{12}})$

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

$10 x^{2} (\sqrt[12]{x^{3}} \sqrt[12]{x^{8}})$

Combine using the product rule for radicals.

$10 x^{2} \sqrt[12]{x^{3} x^{8}}$

Multiply $x^{3}$ by $x^{8}$ by adding the exponents.

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

$10 x^{2} \sqrt[12]{x^{3 + 8}}$

Add $3$ and $8$ .

$10 x^{2} \sqrt[12]{x^{11}}$