Simplify fourth root of 256x^11* square root of 81x^9

$\sqrt[4]{256 x^{11}} \cdot \sqrt{81 x^{9}}$

Step 1

Rewrite $256 x^{11}$ as ${(4 x^{2})}^{4} x^{3}$ .

Rewrite $256$ as $4^{4}$ .

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

Factor out $x^{8}$ .

$\sqrt[4]{4^{4} (x^{8} x^{3})} \cdot \sqrt{81 x^{9}}$

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

$\sqrt[4]{4^{4} ({(x^{2})}^{4} x^{3})} \cdot \sqrt{81 x^{9}}$

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

$\sqrt[4]{{(4 x^{2})}^{4} x^{3}} \cdot \sqrt{81 x^{9}}$

Step 2

Pull terms out from under the radical.

$4 x^{2} \sqrt[4]{x^{3}} \cdot \sqrt{81 x^{9}}$

Step 3

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

Rewrite $81$ as $9^{2}$ .

$4 x^{2} \sqrt[4]{x^{3}} \cdot \sqrt{9^{2} x^{9}}$

Factor out $x^{8}$ .

$4 x^{2} \sqrt[4]{x^{3}} \cdot \sqrt{9^{2} (x^{8} x)}$

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

$4 x^{2} \sqrt[4]{x^{3}} \cdot \sqrt{9^{2} ({(x^{4})}^{2} x)}$

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

$4 x^{2} \sqrt[4]{x^{3}} \cdot \sqrt{{(9 x^{4})}^{2} x}$

Step 4

Pull terms out from under the radical.

$4 x^{2} \sqrt[4]{x^{3}} \cdot (9 x^{4} \sqrt{x})$

Step 5

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$4 (x^{4} x^{2}) \sqrt[4]{x^{3}} \cdot (9 \sqrt{x})$

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

$4 x^{4 + 2} \sqrt[4]{x^{3}} \cdot (9 \sqrt{x})$

Add $4$ and $2$ .

$4 x^{6} \sqrt[4]{x^{3}} \cdot (9 \sqrt{x})$

Step 6

Multiply $4 x^{6} \sqrt[4]{x^{3}} (9 \sqrt{x})$ .

Multiply $9$ by $4$ .

$36 x^{6} \sqrt[4]{x^{3}} \sqrt{x}$

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

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

$36 x^{6} (x^{\frac{1}{2}} \sqrt[4]{x^{3}})$

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

$36 x^{6} (x^{\frac{2}{4}} \sqrt[4]{x^{3}})$

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

$36 x^{6} (\sqrt[4]{x^{2}} \sqrt[4]{x^{3}})$

Combine using the product rule for radicals.

$36 x^{6} \sqrt[4]{x^{2} x^{3}}$

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

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

$36 x^{6} \sqrt[4]{x^{2 + 3}}$

Add $2$ and $3$ .

$36 x^{6} \sqrt[4]{x^{5}}$

Step 7

Factor out $x^{4}$ .

$36 x^{6} \sqrt[4]{x^{4} x}$

Step 8

Pull terms out from under the radical.

$36 x^{6} (x \sqrt[4]{x})$

Step 9

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$36 (x \cdot x^{6}) \sqrt[4]{x}$

Multiply $x$ by $x^{6}$ .

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

$36 (x^{1} x^{6}) \sqrt[4]{x}$

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

$36 x^{1 + 6} \sqrt[4]{x}$

Add $1$ and $6$ .

$36 x^{7} \sqrt[4]{x}$

Simplify fourth root of 256x^11* square root of 81x^9

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