# How do you simplify (sqrt5)/(sqrt5-sqrt3)?

Apr 19, 2018

$\frac{5 + \sqrt{15}}{2}$

#### Explanation:

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

Multiply and divide by $\left(\sqrt{5} + \sqrt{3}\right)$

=> sqrt(5)/(sqrt(5) - sqrt(3)) × (sqrt(5) + sqrt(3))/(sqrt(5) + sqrt(3))

=> (sqrt(5)(sqrt(5) + sqrt(3)))/((sqrt(5) - sqrt(3))(sqrt(5) + sqrt(3))

=> (sqrt(5)(sqrt(5) + sqrt(3)))/((sqrt(5))^2 - (sqrt(3))^2) color(white)(..)[∵ (a - b)(a + b) = a^2 - b^2]

$\implies \frac{\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{3}}{5 - 3}$

$\implies \frac{5 + \sqrt{15}}{2}$

Apr 19, 2018

$\frac{5 + \sqrt{15}}{2}$

#### Explanation:

Multiply (√5) / (√5−√3) by (√5+√3) / (√5+√3) to rationalize the denominator

(√5)/(√5−√3) * (√5+√3) / (√5+√3) = $\frac{\sqrt{5} \cdot \left(\sqrt{5} + \sqrt{3}\right)}{2}$

Apply the distributive property

$\frac{\sqrt{5} \cdot \left(\sqrt{5} + \sqrt{3}\right)}{2}$ = $\frac{\left(\sqrt{5} \cdot \sqrt{5}\right) + \left(\sqrt{5} \cdot \sqrt{3}\right)}{2}$ = $\frac{5 + \sqrt{15}}{2}$

Apr 19, 2018

 = 5/(5 - (sqrt(15))
OR
$= \frac{5}{2} + \frac{\sqrt{15}}{2}$

#### Explanation:

These days, it may be simplest to just use a calculator to complete the expression. But, for purposes of demonstration, we multiply by a radical factor just as we would with another number.
sqrt(5)/(sqrt(5) - sqrt(3)) xx sqrt(5)/(sqrt(5)  = 5/(5 - (sqrt(3) xx sqrt(5))

5/(5 - (sqrt(3) xx sqrt(5)) = 5/(5 - (sqrt(15))

OR
Multiply the denominator and numerator by the same expression as the denominator but with the opposite sign in the middle. This expression is called the conjugate of the denominator.

$\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

$= \frac{5 + \sqrt{15}}{5 - 3}$ = $\frac{5 + \sqrt{15}}{2} = \frac{5}{2} + \frac{\sqrt{15}}{2}$