Simplify 5/( square root of 8- square root of 3)

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

Step 1

Simplify the denominator.

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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

Step 4

Expand the denominator using the FOIL method.

$\frac{5 (2 \sqrt{2} + \sqrt{3})}{4 {\sqrt{2}}^{2} + 2 \sqrt{6} - 2 \sqrt{6} - {\sqrt{3}}^{2}}$

Step 5

Simplify.

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

Step 6

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $2 \sqrt{2} + \sqrt{3}$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$4.56047793 \dots$