Algebra Examples

$\frac{\sqrt{182}}{14 \sqrt{13 + 13 \sqrt{14}}}$

Step 1

Combine $\sqrt{182}$ and $\sqrt{13 + 13 \sqrt{14}}$ into a single radical.

$\frac{\sqrt{\frac{182}{13 + 13 \sqrt{14}}}}{14}$

Step 2

Cancel the common factor of $182$ and $13 + 13 \sqrt{14}$ .

Tap for more steps...

Step 2.1

Factor $13$ out of $182$ .

$\frac{\sqrt{\frac{13 \cdot 14}{13 + 13 \sqrt{14}}}}{14}$

Step 2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1

Factor $13$ out of $13$ .

$\frac{\sqrt{\frac{13 \cdot 14}{13 (1) + 13 \sqrt{14}}}}{14}$

Step 2.2.2

Factor $13$ out of $13 \sqrt{14}$ .

$\frac{\sqrt{\frac{13 \cdot 14}{13 (1) + 13 (\sqrt{14})}}}{14}$

Step 2.2.3

Factor $13$ out of $13 (1) + 13 (\sqrt{14})$ .

$\frac{\sqrt{\frac{13 \cdot 14}{13 (1 + \sqrt{14})}}}{14}$

Step 2.2.4

Cancel the common factor.

$\frac{\sqrt{\frac{13 \cdot 14}{13 (1 + \sqrt{14})}}}{14}$

Step 2.2.5

Rewrite the expression.

$\frac{\sqrt{\frac{14}{1 + \sqrt{14}}}}{14}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Multiply $\frac{14}{1 + \sqrt{14}}$ by $\frac{1 - \sqrt{14}}{1 - \sqrt{14}}$ .

$\frac{\sqrt{\frac{14}{1 + \sqrt{14}} \cdot \frac{1 - \sqrt{14}}{1 - \sqrt{14}}}}{14}$

Step 3.2

Multiply $\frac{14}{1 + \sqrt{14}}$ by $\frac{1 - \sqrt{14}}{1 - \sqrt{14}}$ .

$\frac{\sqrt{\frac{14 (1 - \sqrt{14})}{(1 + \sqrt{14}) (1 - \sqrt{14})}}}{14}$

Step 3.3

Expand the denominator using the FOIL method.

$\frac{\sqrt{\frac{14 (1 - \sqrt{14})}{1 - \sqrt{14} + \sqrt{14} - {\sqrt{14}}^{2}}}}{14}$

Step 3.4

Simplify.

$\frac{\sqrt{\frac{14 (1 - \sqrt{14})}{- 13}}}{14}$

Step 3.5

Move the negative in front of the fraction.

$\frac{\sqrt{- \frac{14 (1 - \sqrt{14})}{13}}}{14}$

Step 3.6

Apply the distributive property.

$\frac{\sqrt{- \frac{14 \cdot 1 + 14 (- \sqrt{14})}{13}}}{14}$

Step 3.7

Multiply $14$ by $1$ .

$\frac{\sqrt{- \frac{14 + 14 (- \sqrt{14})}{13}}}{14}$

Step 3.8

Multiply $- 1$ by $14$ .

$\frac{\sqrt{- \frac{14 - 14 \sqrt{14}}{13}}}{14}$

Step 3.9

Rewrite $- 1$ as $i^{2}$ .

$\frac{\sqrt{i^{2} \frac{14 - 14 \sqrt{14}}{13}}}{14}$

Step 3.10

Pull terms out from under the radical.

$\frac{i \sqrt{\frac{14 - 14 \sqrt{14}}{13}}}{14}$

Step 3.11

Rewrite $\sqrt{\frac{14 - 14 \sqrt{14}}{13}}$ as $\frac{\sqrt{14 - 14 \sqrt{14}}}{\sqrt{13}}$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}}}{\sqrt{13}}}{14}$

Step 3.12

Multiply $\frac{\sqrt{14 - 14 \sqrt{14}}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{i (\frac{\sqrt{14 - 14 \sqrt{14}}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})}{14}$

Step 3.13

Combine and simplify the denominator.

Tap for more steps...

Step 3.13.1

Multiply $\frac{\sqrt{14 - 14 \sqrt{14}}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{\sqrt{13} \sqrt{13}}}{14}$

Step 3.13.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}}{14}$

Step 3.13.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}}{14}$

Step 3.13.4

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

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}}{14}$

Step 3.13.5

Add $1$ and $1$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{{\sqrt{13}}^{2}}}{14}$

Step 3.13.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Tap for more steps...

Step 3.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}}{14}$

Step 3.13.6.2

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

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}}{14}$

Step 3.13.6.3

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

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{13^{\frac{2}{2}}}}{14}$

Step 3.13.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.13.6.4.1

Cancel the common factor.

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{13^{\frac{2}{2}}}}{14}$

Step 3.13.6.4.2

Rewrite the expression.

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{13^{1}}}{14}$

Step 3.13.6.5

Evaluate the exponent.

$\frac{i \frac{\sqrt{14 - 14 \sqrt{14}} \sqrt{13}}{13}}{14}$

Step 3.14

Combine using the product rule for radicals.

$\frac{i \frac{\sqrt{(14 - 14 \sqrt{14}) \cdot 13}}{13}}{14}$

Step 3.15

Combine $i$ and $\frac{\sqrt{(14 - 14 \sqrt{14}) \cdot 13}}{13}$ .

$\frac{\frac{i \sqrt{(14 - 14 \sqrt{14}) \cdot 13}}{13}}{14}$

Step 3.16

Simplify the numerator.

Tap for more steps...

Step 3.16.1

Apply the distributive property.

$\frac{\frac{i \sqrt{14 \cdot 13 - 14 \sqrt{14} \cdot 13}}{13}}{14}$

Step 3.16.2

Multiply $14$ by $13$ .

$\frac{\frac{i \sqrt{182 - 14 \sqrt{14} \cdot 13}}{13}}{14}$

Step 3.16.3

Multiply $13$ by $- 14$ .

$\frac{\frac{i \sqrt{182 - 182 \sqrt{14}}}{13}}{14}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{i \sqrt{182 - 182 \sqrt{14}}}{13} \cdot \frac{1}{14}$

Step 5

Multiply $\frac{i \sqrt{182 - 182 \sqrt{14}}}{13} \cdot \frac{1}{14}$ .

Tap for more steps...

Step 5.1

Multiply $\frac{i \sqrt{182 - 182 \sqrt{14}}}{13}$ by $\frac{1}{14}$ .

$\frac{i \sqrt{182 - 182 \sqrt{14}}}{13 \cdot 14}$

Step 5.2

Multiply $13$ by $14$ .

$\frac{i \sqrt{182 - 182 \sqrt{14}}}{182}$