Basic Math Examples

$\frac{\sqrt{6}}{\sqrt{5} + \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{6}}{\sqrt{5} + \sqrt{2}}$ by $\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

$\frac{\sqrt{6}}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{6}}{\sqrt{5} + \sqrt{2}}$ by $\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{6} (\sqrt{5} - \sqrt{2})}{{\sqrt{5}}^{2} - \sqrt{10} + \sqrt{10} - {\sqrt{2}}^{2}}$

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $6$ by $2$ .

$\frac{\sqrt{6 \cdot 5} - \sqrt{12}}{3}$

Step 8

Simplify each term.

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Step 8.1

Multiply $6$ by $5$ .

$\frac{\sqrt{30} - \sqrt{12}}{3}$

Step 8.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{30} - \sqrt{4 (3)}}{3}$

Step 8.2.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{30} - \sqrt{2^{2} \cdot 3}}{3}$

Step 8.3

Pull terms out from under the radical.

$\frac{\sqrt{30} - (2 \sqrt{3})}{3}$

Step 8.4

Multiply $2$ by $- 1$ .

$\frac{\sqrt{30} - 2 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30} - 2 \sqrt{3}}{3}$

Decimal Form:

$0.67104131 \dots$