Algebra Examples

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

Step 1

Rewrite $\frac{5}{x^{3}}$ as ${(\frac{1}{x})}^{2} \frac{5}{x}$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $5$ .

$2 x^{2} \sqrt{\frac{1^{2} \cdot 5}{x^{3}}}$

Step 1.2

Factor the perfect power $x^{2}$ out of $x^{3}$ .

$2 x^{2} \sqrt{\frac{1^{2} \cdot 5}{x^{2} x}}$

Step 1.3

Rearrange the fraction $\frac{1^{2} \cdot 5}{x^{2} x}$ .

$2 x^{2} \sqrt{{(\frac{1}{x})}^{2} \frac{5}{x}}$

Step 2

Pull terms out from under the radical.

$2 x^{2} (\frac{1}{x} \sqrt{\frac{5}{x}})$

Step 3

Rewrite $\sqrt{\frac{5}{x}}$ as $\frac{\sqrt{5}}{\sqrt{x}}$ .

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5}}{\sqrt{x}})$

Step 4

Multiply $\frac{\sqrt{5}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 x^{2} (\frac{1}{x} (\frac{\sqrt{5}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}))$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{5}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{\sqrt{x} \sqrt{x}})$

Step 5.2

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})$

Step 5.3

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})$

Step 5.4

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1 + 1}})$

Step 5.5

Add $1$ and $1$ .

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{2}})$

Step 5.6

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

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Step 5.6.1

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})$

Step 5.6.2

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})$

Step 5.6.3

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

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 5.6.4.2

Rewrite the expression.

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{x^{1}})$

Step 5.6.5

Simplify.

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5} \sqrt{x}}{x})$

Step 6

Combine using the product rule for radicals.

$2 x^{2} (\frac{1}{x} \cdot \frac{\sqrt{5 x}}{x})$

Step 7

Multiply $\frac{1}{x} \cdot \frac{\sqrt{5 x}}{x}$ .

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Step 7.1

Multiply $\frac{1}{x}$ by $\frac{\sqrt{5 x}}{x}$ .

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

Step 7.2

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

$2 x^{2} \frac{\sqrt{5 x}}{x^{1} x}$

Step 7.3

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

$2 x^{2} \frac{\sqrt{5 x}}{x^{1} x^{1}}$

Step 7.4

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

$2 x^{2} \frac{\sqrt{5 x}}{x^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

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

Step 8

Cancel the common factor of $x^{2}$ .

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Step 8.1

Factor $x^{2}$ out of $2 x^{2}$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

$2 \sqrt{5 x}$