Question

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

Answer 1

`(5 + sqrt(15))/2`

Explanation:

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

Multiply and divide by `(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) - 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]`

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

`=> (5 + sqrt(15))/2`

Answer 2

`(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)` = `(sqrt5*(sqrt5 + sqrt3)) / 2`

Apply the distributive property

`(sqrt5*(sqrt5 + sqrt3)) / 2` = `((sqrt5*sqrt5)+(sqrt5*sqrt3))/2` = `(5+sqrt(15)) / 2`

Answer 3

`= 5/(5 - (sqrt(15))`
OR
`= 5/2 + sqrt(15)/2`
Take your pick.

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.

`sqrt(5)/(sqrt(5) - sqrt(3)) xx (sqrt(5) + sqrt(3))/(sqrt(5) + sqrt(3))`

`= (5 + sqrt(15))/(5 - 3)` = `(5 + sqrt(15))/2 = 5/2 + sqrt(15)/2`

https://www.mathportal.org/algebra/roots-and-radicals/multiplying-and-dividing-radicals.php