Algebra Examples

$\frac{x^{2}}{\sqrt{12 x^{3}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12 x^{3}$ as ${(2 x)}^{2} \cdot (3 x)$ .

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.3

Factor out $x^{2}$ .

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

Step 1.1.4

Move $3$ .

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

Step 1.1.5

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

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

Step 1.1.6

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $x$ out of $2 x \sqrt{3 x}$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Multiply $\frac{x}{2 \sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{x}{2 \sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

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

Step 4.1.2

Move $\sqrt{3 x}$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Add $1$ and $1$ .

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

Step 4.1.7

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

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Step 4.1.7.1

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

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

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

Step 4.1.7.4.2

Rewrite the expression.

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

Step 4.1.7.5

Simplify.

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 5

Multiply $2$ by $3$ .

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