Algebra Examples

$\sqrt[3]{64 x} \cdot \sqrt[4]{256 x^{2}}$

Step 1

Rewrite $64$ as $4^{3}$ .

$\sqrt[3]{4^{3} x} \cdot \sqrt[4]{256 x^{2}}$

Step 2

Pull terms out from under the radical.

$4 \sqrt[3]{x} \cdot \sqrt[4]{256 x^{2}}$

Step 3

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

Multiply $4$ by $4$ .

$16 \sqrt[3]{x} \sqrt{x}$

Step 7.2

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

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Step 7.2.1

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

$16 (x^{\frac{1}{2}} \sqrt[3]{x})$

Step 7.2.2

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

$16 (x^{\frac{3}{6}} \sqrt[3]{x})$

Step 7.2.3

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

$16 (\sqrt[6]{x^{3}} \sqrt[3]{x})$

Step 7.2.4

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

$16 (\sqrt[6]{x^{3}} x^{\frac{1}{3}})$

Step 7.2.5

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

$16 (\sqrt[6]{x^{3}} x^{\frac{2}{6}})$

Step 7.2.6

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

$16 (\sqrt[6]{x^{3}} \sqrt[6]{x^{2}})$

Step 7.3

Combine using the product rule for radicals.

$16 \sqrt[6]{x^{3} x^{2}}$

Step 7.4

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

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Step 7.4.1

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

$16 \sqrt[6]{x^{3 + 2}}$

Step 7.4.2

Add $3$ and $2$ .

$16 \sqrt[6]{x^{5}}$