Write the Fraction in Simplest Form ( square root of 6)/( square root of 3- square root of 5)

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

Step 1

Multiply $\frac{\sqrt{6}}{\sqrt{3} - \sqrt{5}}$ by $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}}$ .

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

Step 2

Multiply $\frac{\sqrt{6}}{\sqrt{3} - \sqrt{5}}$ by $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{6} (\sqrt{3} + \sqrt{5})}{{\sqrt{3}}^{2} + \sqrt{15} - \sqrt{15} - {\sqrt{5}}^{2}}$

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

Combine using the product rule for radicals.

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

Step 8

Simplify each term.

Multiply $6$ by $3$ .

$\frac{\sqrt{18} + \sqrt{6 \cdot 5}}{- 2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} + \sqrt{6 \cdot 5}}{- 2}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $6$ by $5$ .

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

Step 9

Move the negative in front of the fraction.

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