Algebra Examples

$\frac{1}{\sqrt{12 z}}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$\frac{1}{\sqrt{4 (3) z}}$

Step 1.1.2

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

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{1}{2 \sqrt{3 z}}$ by $\frac{\sqrt{3 z}}{\sqrt{3 z}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{2 \sqrt{3 z}}$ by $\frac{\sqrt{3 z}}{\sqrt{3 z}}$ .

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

Step 3.2

Move $\sqrt{3 z}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Simplify.

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

Step 4

Multiply $2$ by $3$ .

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