Algebra Examples

$\frac{3}{\sqrt{8} + \sqrt{5}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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

Step 4

Expand the denominator using the FOIL method.

$\frac{3 (2 \sqrt{2} - \sqrt{5})}{4 {\sqrt{2}}^{2} - 2 \sqrt{10} + 2 \sqrt{10} - {\sqrt{5}}^{2}}$

Step 5

Simplify.

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

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.59235914 \dots$