Algebra Examples

$\frac{2 \sqrt{x} \cdot \sqrt{x^{3}}}{\sqrt{9 x^{10}}}$

Step 1

Combine $\sqrt{x}$ and $\sqrt{9 x^{10}}$ into a single radical.

$2 \sqrt{\frac{x}{9 x^{10}}} \cdot \sqrt{x^{3}}$

Step 2

Reduce the expression $\frac{x}{9 x^{10}}$ by cancelling the common factors.

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Step 2.1

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

$2 \sqrt{\frac{x^{1}}{9 x^{10}}} \cdot \sqrt{x^{3}}$

Step 2.2

Factor $x$ out of $x^{1}$ .

$2 \sqrt{\frac{x \cdot 1}{9 x^{10}}} \cdot \sqrt{x^{3}}$

Step 2.3

Factor $x$ out of $9 x^{10}$ .

$2 \sqrt{\frac{x \cdot 1}{x (9 x^{9})}} \cdot \sqrt{x^{3}}$

Step 2.4

Cancel the common factor.

$2 \sqrt{\frac{x \cdot 1}{x (9 x^{9})}} \cdot \sqrt{x^{3}}$

Step 2.5

Rewrite the expression.

$2 \sqrt{\frac{1}{9 x^{9}}} \cdot \sqrt{x^{3}}$

Step 3

Rewrite $\frac{1}{9 x^{9}}$ as ${(\frac{1}{3 x^{4}})}^{2} \frac{1}{x}$ .

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Step 3.1

Factor the perfect power $1^{2}$ out of $1$ .

$2 \sqrt{\frac{1^{2} \cdot 1}{9 x^{9}}} \cdot \sqrt{x^{3}}$

Step 3.2

Factor the perfect power ${(3 x^{4})}^{2}$ out of $9 x^{9}$ .

$2 \sqrt{\frac{1^{2} \cdot 1}{{(3 x^{4})}^{2} x}} \cdot \sqrt{x^{3}}$

Step 3.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{{(3 x^{4})}^{2} x}$ .

$2 \sqrt{{(\frac{1}{3 x^{4}})}^{2} \frac{1}{x}} \cdot \sqrt{x^{3}}$

Step 4

Pull terms out from under the radical.

$2 (\frac{1}{3 x^{4}} \sqrt{\frac{1}{x}}) \cdot \sqrt{x^{3}}$

Step 5

Rewrite $\sqrt{\frac{1}{x}}$ as $\frac{\sqrt{1}}{\sqrt{x}}$ .

$2 (\frac{1}{3 x^{4}} \cdot \frac{\sqrt{1}}{\sqrt{x}}) \cdot \sqrt{x^{3}}$

Step 6

Combine.

$2 \frac{1 \sqrt{1}}{3 x^{4} \sqrt{x}} \cdot \sqrt{x^{3}}$

Step 7

Multiply $\sqrt{1}$ by $1$ .

$2 \frac{\sqrt{1}}{3 x^{4} \sqrt{x}} \cdot \sqrt{x^{3}}$

Step 8

Any root of $1$ is $1$ .

$2 \frac{1}{3 x^{4} \sqrt{x}} \cdot \sqrt{x^{3}}$

Step 9

Multiply $\frac{1}{3 x^{4} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 (\frac{1}{3 x^{4} \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) \cdot \sqrt{x^{3}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{3 x^{4} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 \frac{\sqrt{x}}{3 x^{4} \sqrt{x} \sqrt{x}} \cdot \sqrt{x^{3}}$

Step 10.2

Move $\sqrt{x}$ .

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

Step 10.3

Raise $\sqrt{x}$ to the power of $1$ .

$2 \frac{\sqrt{x}}{3 x^{4} ({\sqrt{x}}^{1} \sqrt{x})} \cdot \sqrt{x^{3}}$

Step 10.4

Raise $\sqrt{x}$ to the power of $1$ .

$2 \frac{\sqrt{x}}{3 x^{4} ({\sqrt{x}}^{1} {\sqrt{x}}^{1})} \cdot \sqrt{x^{3}}$

Step 10.5

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

$2 \frac{\sqrt{x}}{3 x^{4} {\sqrt{x}}^{1 + 1}} \cdot \sqrt{x^{3}}$

Step 10.6

Add $1$ and $1$ .

$2 \frac{\sqrt{x}}{3 x^{4} {\sqrt{x}}^{2}} \cdot \sqrt{x^{3}}$

Step 10.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 10.7.1

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

$2 \frac{\sqrt{x}}{3 x^{4} {(x^{\frac{1}{2}})}^{2}} \cdot \sqrt{x^{3}}$

Step 10.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \frac{\sqrt{x}}{3 x^{4} x^{\frac{1}{2} \cdot 2}} \cdot \sqrt{x^{3}}$

Step 10.7.3

Combine $\frac{1}{2}$ and $2$ .

$2 \frac{\sqrt{x}}{3 x^{4} x^{\frac{2}{2}}} \cdot \sqrt{x^{3}}$

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

$2 \frac{\sqrt{x}}{3 x^{4} x^{\frac{2}{2}}} \cdot \sqrt{x^{3}}$

Step 10.7.4.2

Rewrite the expression.

$2 \frac{\sqrt{x}}{3 x^{4} x^{1}} \cdot \sqrt{x^{3}}$

Step 10.7.5

Simplify.

$2 \frac{\sqrt{x}}{3 x^{4} x} \cdot \sqrt{x^{3}}$

Step 11

Simplify the denominator.

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Step 11.1

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

$2 \frac{\sqrt{x}}{3 (x^{1} x^{4})} \cdot \sqrt{x^{3}}$

Step 11.2

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

$2 \frac{\sqrt{x}}{3 x^{1 + 4}} \cdot \sqrt{x^{3}}$

Step 11.3

Add $1$ and $4$ .

$2 \frac{\sqrt{x}}{3 x^{5}} \cdot \sqrt{x^{3}}$

Step 12

Combine $2$ and $\frac{\sqrt{x}}{3 x^{5}}$ .

$\frac{2 \sqrt{x}}{3 x^{5}} \cdot \sqrt{x^{3}}$

Step 13

Factor out $x^{2}$ .

$\frac{2 \sqrt{x}}{3 x^{5}} \cdot \sqrt{x^{2} x}$

Step 14

Pull terms out from under the radical.

$\frac{2 \sqrt{x}}{3 x^{5}} \cdot (x \sqrt{x})$

Step 15

Cancel the common factor of $x$ .

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Step 15.1

Factor $x$ out of $3 x^{5}$ .

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

Step 15.2

Factor $x$ out of $x \sqrt{x}$ .

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

Step 15.3

Cancel the common factor.

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

Step 15.4

Rewrite the expression.

$\frac{2 \sqrt{x}}{3 x^{4}} \cdot \sqrt{x}$

Step 16

Combine $\frac{2 \sqrt{x}}{3 x^{4}}$ and $\sqrt{x}$ .

$\frac{2 \sqrt{x} \sqrt{x}}{3 x^{4}}$

Step 17

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{2 ({\sqrt{x}}^{1} \sqrt{x})}{3 x^{4}}$

Step 18

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{2 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}{3 x^{4}}$

Step 19

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

$\frac{2 {\sqrt{x}}^{1 + 1}}{3 x^{4}}$

Step 20

Add $1$ and $1$ .

$\frac{2 {\sqrt{x}}^{2}}{3 x^{4}}$

Step 21

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 21.1

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

$\frac{2 {(x^{\frac{1}{2}})}^{2}}{3 x^{4}}$

Step 21.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 x^{\frac{1}{2} \cdot 2}}{3 x^{4}}$

Step 21.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 x^{\frac{2}{2}}}{3 x^{4}}$

Step 21.4

Cancel the common factor of $2$ .

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Step 21.4.1

Cancel the common factor.

$\frac{2 x^{\frac{2}{2}}}{3 x^{4}}$

Step 21.4.2

Rewrite the expression.

$\frac{2 x^{1}}{3 x^{4}}$

Step 21.5

Simplify.

$\frac{2 x}{3 x^{4}}$

Step 22

Cancel the common factor of $x$ and $x^{4}$ .

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Step 22.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{3 x^{4}}$

Step 22.2

Cancel the common factors.

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Step 22.2.1

Factor $x$ out of $3 x^{4}$ .

$\frac{x \cdot 2}{x (3 x^{3})}$

Step 22.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x (3 x^{3})}$

Step 22.2.3

Rewrite the expression.

$\frac{2}{3 x^{3}}$