Algebra Examples

$\frac{\sqrt{5 x}}{2 \sqrt{15 x^{4}}}$

Step 1

Combine $\sqrt{5 x}$ and $\sqrt{15 x^{4}}$ into a single radical.

$\frac{\sqrt{\frac{5 x}{15 x^{4}}}}{2}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $5$ and $15$ .

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Step 2.1.1

Factor $5$ out of $5 x$ .

$\frac{\sqrt{\frac{5 (x)}{15 x^{4}}}}{2}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $5$ out of $15 x^{4}$ .

$\frac{\sqrt{\frac{5 (x)}{5 (3 x^{4})}}}{2}$

Step 2.1.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{5 x}{5 (3 x^{4})}}}{2}$

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Cancel the common factors.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Cancel the common factor.

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

Step 2.2.3.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Pull terms out from under the radical.

$\frac{\frac{1}{x} \sqrt{\frac{1}{3 x}}}{2}$

Step 3.3

Rewrite $\sqrt{\frac{1}{3 x}}$ as $\frac{\sqrt{1}}{\sqrt{3 x}}$ .

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

Step 3.4

Any root of $1$ is $1$ .

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

Step 3.5

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

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

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

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

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

Step 3.6.2

Raise $\sqrt{3 x}$ to the power of $1$ .

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

Step 3.6.3

Raise $\sqrt{3 x}$ to the power of $1$ .

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

Step 3.6.4

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

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

Step 3.6.5

Add $1$ and $1$ .

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

Step 3.6.6

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Step 3.6.6.2

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

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

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $2$ .

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Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

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

Step 3.6.6.5

Simplify.

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

Step 3.7

Multiply $\frac{1}{x} \cdot \frac{\sqrt{3 x}}{3 x}$ .

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Step 3.7.1

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

$\frac{\frac{\sqrt{3 x}}{x (3 x)}}{2}$

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

Add $1$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply.

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Step 6.1

Multiply $\sqrt{3 x}$ by $1$ .

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

Step 6.2

Multiply $2$ by $3$ .

$\frac{\sqrt{3 x}}{6 x^{2}}$