Algebra Examples

$\sqrt{\frac{3}{5 - \sqrt{6}}}$

Step 1

Multiply $\frac{3}{5 - \sqrt{6}}$ by $\frac{5 + \sqrt{6}}{5 + \sqrt{6}}$ .

$\sqrt{\frac{3}{5 - \sqrt{6}} \cdot \frac{5 + \sqrt{6}}{5 + \sqrt{6}}}$

Step 2

Multiply $\frac{3}{5 - \sqrt{6}}$ by $\frac{5 + \sqrt{6}}{5 + \sqrt{6}}$ .

$\sqrt{\frac{3 (5 + \sqrt{6})}{(5 - \sqrt{6}) (5 + \sqrt{6})}}$

Step 3

Expand the denominator using the FOIL method.

$\sqrt{\frac{3 (5 + \sqrt{6})}{25 + 5 \sqrt{6} - 5 \sqrt{6} - {\sqrt{6}}^{2}}}$

Step 4

Simplify.

$\sqrt{\frac{3 (5 + \sqrt{6})}{19}}$

Step 5

Rewrite $\sqrt{\frac{3 (5 + \sqrt{6})}{19}}$ as $\frac{\sqrt{3 (5 + \sqrt{6})}}{\sqrt{19}}$ .

$\frac{\sqrt{3 (5 + \sqrt{6})}}{\sqrt{19}}$

Step 6

Multiply $\frac{\sqrt{3 (5 + \sqrt{6})}}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\frac{\sqrt{3 (5 + \sqrt{6})}}{\sqrt{19}} \cdot \frac{\sqrt{19}}{\sqrt{19}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{3 (5 + \sqrt{6})}}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{\sqrt{19} \sqrt{19}}$

Step 7.2

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{{\sqrt{19}}^{1} \sqrt{19}}$

Step 7.3

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{{\sqrt{19}}^{1} {\sqrt{19}}^{1}}$

Step 7.4

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

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{{\sqrt{19}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{{\sqrt{19}}^{2}}$

Step 7.6

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{{(19^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{19^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{19^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{19^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{19^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{3 (5 + \sqrt{6})} \sqrt{19}}{19}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 (5 + \sqrt{6}) \cdot 19}}{19}$

Step 8.2

Multiply $19$ by $3$ .

$\frac{\sqrt{57 (5 + \sqrt{6})}}{19}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{57 (5 + \sqrt{6})}}{19}$

Decimal Form:

$1.08454378 \dots$