Simplify square root of 121x^9* fourth root of 81x

$\sqrt{121 x^{9}} \cdot \sqrt[4]{81 x}$

Step 1

Rewrite $121 x^{9}$ as ${(11 x^{4})}^{2} x$ .

Rewrite $121$ as $11^{2}$ .

$\sqrt{11^{2} x^{9}} \cdot \sqrt[4]{81 x}$

Factor out $x^{8}$ .

$\sqrt{11^{2} (x^{8} x)} \cdot \sqrt[4]{81 x}$

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$\sqrt{11^{2} ({(x^{4})}^{2} x)} \cdot \sqrt[4]{81 x}$

Rewrite $11^{2} {(x^{4})}^{2}$ as ${(11 x^{4})}^{2}$ .

$\sqrt{{(11 x^{4})}^{2} x} \cdot \sqrt[4]{81 x}$

Step 2

Pull terms out from under the radical.

$11 x^{4} \sqrt{x} \cdot \sqrt[4]{81 x}$

Step 3

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

$11 x^{4} \sqrt{x} \cdot \sqrt[4]{3^{4} x}$

Step 4

Pull terms out from under the radical.

$11 x^{4} \sqrt{x} \cdot (3 \sqrt[4]{x})$

Step 5

Multiply $11 x^{4} \sqrt{x} (3 \sqrt[4]{x})$ .

Multiply $3$ by $11$ .

$33 x^{4} \sqrt{x} \sqrt[4]{x}$

Rewrite the expression using the least common index of $4$ .

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

$33 x^{4} (\sqrt[4]{x} x^{\frac{1}{2}})$

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

$33 x^{4} (\sqrt[4]{x} x^{\frac{2}{4}})$

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

$33 x^{4} (\sqrt[4]{x} \sqrt[4]{x^{2}})$

Combine using the product rule for radicals.

$33 x^{4} \sqrt[4]{x \cdot x^{2}}$

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

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$33 x^{4} \sqrt[4]{x^{1} x^{2}}$

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

$33 x^{4} \sqrt[4]{x^{1 + 2}}$

Add $1$ and $2$ .

$33 x^{4} \sqrt[4]{x^{3}}$