Simplify fourth root of 81x^9* square root of 144x

$\sqrt[4]{81 x^{9}} \cdot \sqrt{144 x}$

Step 1

Rewrite $81 x^{9}$ as ${(3 x^{2})}^{4} x$ .

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

$\sqrt[4]{3^{4} x^{9}} \cdot \sqrt{144 x}$

Factor out $x^{8}$ .

$\sqrt[4]{3^{4} (x^{8} x)} \cdot \sqrt{144 x}$

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

$\sqrt[4]{3^{4} ({(x^{2})}^{4} x)} \cdot \sqrt{144 x}$

Rewrite $3^{4} {(x^{2})}^{4}$ as ${(3 x^{2})}^{4}$ .

$\sqrt[4]{{(3 x^{2})}^{4} x} \cdot \sqrt{144 x}$

Step 2

Pull terms out from under the radical.

$3 x^{2} \sqrt[4]{x} \cdot \sqrt{144 x}$

Step 3

Rewrite $144$ as $12^{2}$ .

$3 x^{2} \sqrt[4]{x} \cdot \sqrt{12^{2} x}$

Step 4

Pull terms out from under the radical.

$3 x^{2} \sqrt[4]{x} \cdot (12 \sqrt{x})$

Step 5

Multiply $3 x^{2} \sqrt[4]{x} (12 \sqrt{x})$ .

Multiply $12$ by $3$ .

$36 x^{2} \sqrt[4]{x} \sqrt{x}$

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

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

$36 x^{2} (x^{\frac{1}{2}} \sqrt[4]{x})$

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

$36 x^{2} (x^{\frac{2}{4}} \sqrt[4]{x})$

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

$36 x^{2} (\sqrt[4]{x^{2}} \sqrt[4]{x})$

Combine using the product rule for radicals.

$36 x^{2} \sqrt[4]{x^{2} x}$

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

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

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

$36 x^{2} \sqrt[4]{x^{2} x^{1}}$

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

$36 x^{2} \sqrt[4]{x^{2 + 1}}$

Add $2$ and $1$ .

$36 x^{2} \sqrt[4]{x^{3}}$