Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{\sqrt{5} (5 - \sqrt{2})}{23}$

Step 5

Apply the distributive property.

$\frac{\sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{2})}{23}$

Step 6

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

$\frac{5 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{2})}{23}$

Step 7

Multiply $\sqrt{5} (- \sqrt{2})$ .

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Step 7.1

Combine using the product rule for radicals.

$\frac{5 \cdot \sqrt{5} - \sqrt{5 \cdot 2}}{23}$

Step 7.2

Multiply $5$ by $2$ .

$\frac{5 \cdot \sqrt{5} - \sqrt{10}}{23}$

$\frac{5 \sqrt{5} - \sqrt{10}}{23}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{5} - \sqrt{10}}{23}$

Decimal Form:

$0.34861140 \dots$