Simplify fourth root of 256x^9* square root of 64x^11

$\sqrt[4]{256 x^{9}} \cdot \sqrt{64 x^{11}}$

Step 1

Rewrite $256 x^{9}$ as ${(4 x^{2})}^{4} x$ .

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

$\sqrt[4]{4^{4} x^{9}} \cdot \sqrt{64 x^{11}}$

Factor out $x^{8}$ .

$\sqrt[4]{4^{4} (x^{8} x)} \cdot \sqrt{64 x^{11}}$

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

$\sqrt[4]{4^{4} ({(x^{2})}^{4} x)} \cdot \sqrt{64 x^{11}}$

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

$\sqrt[4]{{(4 x^{2})}^{4} x} \cdot \sqrt{64 x^{11}}$

Step 2

Pull terms out from under the radical.

$4 x^{2} \sqrt[4]{x} \cdot \sqrt{64 x^{11}}$

Step 3

Rewrite $64 x^{11}$ as ${(8 x^{5})}^{2} x$ .

Rewrite $64$ as $8^{2}$ .

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

Factor out $x^{10}$ .

$4 x^{2} \sqrt[4]{x} \cdot \sqrt{8^{2} (x^{10} x)}$

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

$4 x^{2} \sqrt[4]{x} \cdot \sqrt{8^{2} ({(x^{5})}^{2} x)}$

Rewrite $8^{2} {(x^{5})}^{2}$ as ${(8 x^{5})}^{2}$ .

$4 x^{2} \sqrt[4]{x} \cdot \sqrt{{(8 x^{5})}^{2} x}$

Step 4

Pull terms out from under the radical.

$4 x^{2} \sqrt[4]{x} \cdot (8 x^{5} \sqrt{x})$

Step 5

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

Move $x^{5}$ .

$4 (x^{5} x^{2}) \sqrt[4]{x} \cdot (8 \sqrt{x})$

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

$4 x^{5 + 2} \sqrt[4]{x} \cdot (8 \sqrt{x})$

Add $5$ and $2$ .

$4 x^{7} \sqrt[4]{x} \cdot (8 \sqrt{x})$

Step 6

Multiply $4 x^{7} \sqrt[4]{x} (8 \sqrt{x})$ .

Multiply $8$ by $4$ .

$32 x^{7} \sqrt[4]{x} \sqrt{x}$

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

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

$32 x^{7} (x^{\frac{1}{2}} \sqrt[4]{x})$

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

$32 x^{7} (x^{\frac{2}{4}} \sqrt[4]{x})$

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

$32 x^{7} (\sqrt[4]{x^{2}} \sqrt[4]{x})$

Combine using the product rule for radicals.

$32 x^{7} \sqrt[4]{x^{2} x}$

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

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

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

$32 x^{7} \sqrt[4]{x^{2} x^{1}}$

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

$32 x^{7} \sqrt[4]{x^{2 + 1}}$

Add $2$ and $1$ .

$32 x^{7} \sqrt[4]{x^{3}}$

Simplify fourth root of 256x^9* square root of 64x^11

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