Simplify fourth root of 81x^2* cube root of 27x^7

$\sqrt[4]{81 x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Rewrite $81$ as $3^{4}$ .

$\sqrt[4]{3^{4} x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Pull terms out from under the radical.

$3 \sqrt[4]{x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$3 \sqrt{\sqrt{x^{2}}} \cdot \sqrt[3]{27 x^{7}}$

Pull terms out from under the radical, assuming positive real numbers.

$3 \sqrt{x} \cdot \sqrt[3]{27 x^{7}}$

Rewrite $27 x^{7}$ as ${(3 x^{2})}^{3} x$ .

Rewrite $27$ as $3^{3}$ .

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

Factor out $x^{6}$ .

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

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

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

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

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

Pull terms out from under the radical.

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

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

Multiply $3$ by $3$ .

$9 \sqrt{x} (x^{2} \sqrt[3]{x})$

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

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

$9 (x^{2} (x^{\frac{1}{2}} \sqrt[3]{x}))$

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

$9 (x^{2} (x^{\frac{3}{6}} \sqrt[3]{x}))$

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

$9 (x^{2} (\sqrt[6]{x^{3}} \sqrt[3]{x}))$

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

$9 (x^{2} (\sqrt[6]{x^{3}} x^{\frac{1}{3}}))$

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

$9 (x^{2} (\sqrt[6]{x^{3}} x^{\frac{2}{6}}))$

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

$9 (x^{2} (\sqrt[6]{x^{3}} \sqrt[6]{x^{2}}))$

Combine using the product rule for radicals.

$9 (x^{2} \sqrt[6]{x^{3} x^{2}})$

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

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

$9 (x^{2} \sqrt[6]{x^{3 + 2}})$

Add $3$ and $2$ .

$9 (x^{2} \sqrt[6]{x^{5}})$

$9 x^{2} \sqrt[6]{x^{5}}$