Algebra Examples

$\sqrt{100 x^{3}} \cdot \sqrt[3]{64 x^{10}}$

Step 1

Rewrite $100 x^{3}$ as ${(10 x)}^{2} x$ .

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Step 1.1

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2} x^{3}} \cdot \sqrt[3]{64 x^{10}}$

Step 1.2

Factor out $x^{2}$ .

$\sqrt{10^{2} (x^{2} x)} \cdot \sqrt[3]{64 x^{10}}$

Step 1.3

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

$\sqrt{{(10 x)}^{2} x} \cdot \sqrt[3]{64 x^{10}}$

Step 2

Pull terms out from under the radical.

$10 x \sqrt{x} \cdot \sqrt[3]{64 x^{10}}$

Step 3

Rewrite $64 x^{10}$ as ${(4 x^{3})}^{3} x$ .

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Step 3.1

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

$10 x \sqrt{x} \cdot \sqrt[3]{4^{3} x^{10}}$

Step 3.2

Factor out $x^{9}$ .

$10 x \sqrt{x} \cdot \sqrt[3]{4^{3} (x^{9} x)}$

Step 3.3

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

$10 x \sqrt{x} \cdot \sqrt[3]{4^{3} ({(x^{3})}^{3} x)}$

Step 3.4

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

$10 x \sqrt{x} \cdot \sqrt[3]{{(4 x^{3})}^{3} x}$

Step 4

Pull terms out from under the radical.

$10 x \sqrt{x} \cdot (4 x^{3} \sqrt[3]{x})$

Step 5

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

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Step 5.1

Move $x^{3}$ .

$10 (x^{3} x) \sqrt{x} \cdot (4 \sqrt[3]{x})$

Step 5.2

Multiply $x^{3}$ by $x$ .

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Step 5.2.1

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

$10 (x^{3} x^{1}) \sqrt{x} \cdot (4 \sqrt[3]{x})$

Step 5.2.2

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

$10 x^{3 + 1} \sqrt{x} \cdot (4 \sqrt[3]{x})$

Step 5.3

Add $3$ and $1$ .

$10 x^{4} \sqrt{x} \cdot (4 \sqrt[3]{x})$

Step 6

Multiply $10 x^{4} \sqrt{x} (4 \sqrt[3]{x})$ .

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Step 6.1

Multiply $4$ by $10$ .

$40 x^{4} \sqrt{x} \sqrt[3]{x}$

Step 6.2

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

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Step 6.2.1

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

$40 x^{4} (x^{\frac{1}{3}} \sqrt{x})$

Step 6.2.2

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

$40 x^{4} (x^{\frac{2}{6}} \sqrt{x})$

Step 6.2.3

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

$40 x^{4} (\sqrt[6]{x^{2}} \sqrt{x})$

Step 6.2.4

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

$40 x^{4} (\sqrt[6]{x^{2}} x^{\frac{1}{2}})$

Step 6.2.5

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

$40 x^{4} (\sqrt[6]{x^{2}} x^{\frac{3}{6}})$

Step 6.2.6

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

$40 x^{4} (\sqrt[6]{x^{2}} \sqrt[6]{x^{3}})$

Step 6.3

Combine using the product rule for radicals.

$40 x^{4} \sqrt[6]{x^{2} x^{3}}$

Step 6.4

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

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Step 6.4.1

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

$40 x^{4} \sqrt[6]{x^{2 + 3}}$

Step 6.4.2

Add $2$ and $3$ .

$40 x^{4} \sqrt[6]{x^{5}}$