Simplify square root of 25x^11* cube root of 216x

$\sqrt{25 x^{11}} \cdot \sqrt[3]{216 x}$

Rewrite $25 x^{11}$ as ${(5 x^{5})}^{2} x$ .

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} x^{11}} \cdot \sqrt[3]{216 x}$

Factor out $x^{10}$ .

$\sqrt{5^{2} (x^{10} x)} \cdot \sqrt[3]{216 x}$

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

$\sqrt{5^{2} ({(x^{5})}^{2} x)} \cdot \sqrt[3]{216 x}$

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

$\sqrt{{(5 x^{5})}^{2} x} \cdot \sqrt[3]{216 x}$

Pull terms out from under the radical.

$5 x^{5} \sqrt{x} \cdot \sqrt[3]{216 x}$

Rewrite $216$ as $6^{3}$ .

$5 x^{5} \sqrt{x} \cdot \sqrt[3]{6^{3} x}$

Pull terms out from under the radical.

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

Multiply $5 x^{5} \sqrt{x} (6 \sqrt[3]{x})$ .

Multiply $6$ by $5$ .

$30 x^{5} \sqrt{x} \sqrt[3]{x}$

Rewrite the expression using the least common index of $6$ .

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

$30 x^{5} (x^{\frac{1}{3}} \sqrt{x})$

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$30 x^{5} (x^{\frac{2}{6}} \sqrt{x})$

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$30 x^{5} (\sqrt[6]{x^{2}} \sqrt{x})$

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

$30 x^{5} (\sqrt[6]{x^{2}} x^{\frac{1}{2}})$

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$30 x^{5} (\sqrt[6]{x^{2}} x^{\frac{3}{6}})$

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$30 x^{5} (\sqrt[6]{x^{2}} \sqrt[6]{x^{3}})$

Combine using the product rule for radicals.

$30 x^{5} \sqrt[6]{x^{2} x^{3}}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

$30 x^{5} \sqrt[6]{x^{2 + 3}}$

Add $2$ and $3$ .

$30 x^{5} \sqrt[6]{x^{5}}$