Algebra Examples

$\frac{\sqrt{5}}{2 + \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{5}}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$\frac{\sqrt{5}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$

Step 2

Multiply $\frac{\sqrt{5}}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} (2 - \sqrt{3})}{4 - 2 \sqrt{3} + \sqrt{3} \cdot 2 - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{5} (2 - \sqrt{3})}{1}$

Step 5

Divide $\sqrt{5} (2 - \sqrt{3})$ by $1$ .

$\sqrt{5} (2 - \sqrt{3})$

Step 6

Apply the distributive property.

$\sqrt{5} \cdot 2 + \sqrt{5} (- \sqrt{3})$

Step 7

Move $2$ to the left of $\sqrt{5}$ .

$2 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{3})$

Step 8

Multiply $\sqrt{5} (- \sqrt{3})$ .

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Step 8.1

Combine using the product rule for radicals.

$2 \cdot \sqrt{5} - \sqrt{5 \cdot 3}$

Step 8.2

Multiply $5$ by $3$ .

$2 \cdot \sqrt{5} - \sqrt{15}$

$2 \sqrt{5} - \sqrt{15}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{5} - \sqrt{15}$

Decimal Form:

$0.59915260 \dots$