Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{\sqrt{6} (5 - \sqrt{3})}{22}$

Step 5

Apply the distributive property.

$\frac{\sqrt{6} \cdot 5 + \sqrt{6} (- \sqrt{3})}{22}$

Step 6

Move $5$ to the left of $\sqrt{6}$ .

$\frac{5 \cdot \sqrt{6} + \sqrt{6} (- \sqrt{3})}{22}$

Step 7

Multiply $\sqrt{6} (- \sqrt{3})$ .

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Step 7.1

Combine using the product rule for radicals.

$\frac{5 \cdot \sqrt{6} - \sqrt{6 \cdot 3}}{22}$

Step 7.2

Multiply $6$ by $3$ .

$\frac{5 \cdot \sqrt{6} - \sqrt{18}}{22}$

Step 8

Simplify each term.

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Step 8.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 8.1.1

Factor $9$ out of $18$ .

$\frac{5 \sqrt{6} - \sqrt{9 (2)}}{22}$

Step 8.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{5 \sqrt{6} - \sqrt{3^{2} \cdot 2}}{22}$

Step 8.2

Pull terms out from under the radical.

$\frac{5 \sqrt{6} - (3 \sqrt{2})}{22}$

Step 8.3

Multiply $3$ by $- 1$ .

$\frac{5 \sqrt{6} - 3 \sqrt{2}}{22}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{6} - 3 \sqrt{2}}{22}$

Decimal Form:

$0.36385491 \dots$