Question

Simplify the following:

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

Answer 1

We are tasked with simplifying the following expression:

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

Step 1: Rationalise both fractions

We’ll begin by rationalising the denominators of both fractions by multiplying the numerator and denominator by the conjugate of the denominator.

First term: `(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))`

Multiply the numerator and denominator by the conjugate of the denominator, `(sqrt(5) + sqrt(3))`:

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

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

Simplify the denominator:

`(sqrt(5))^2 - (sqrt(3))^2`

= 5 – 3

= 2

Now expand the numerator:

`(sqrt(5) + sqrt(3))^2`

= `(sqrt(5))^2 + 2sqrt(5)sqrt(3) + (sqrt(3))^2`

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

= `8 + 2sqrt(15)`

So the first term becomes:

`(8 + 2sqrt(15))/2 = 4 + sqrt(15)`

Second term: `(sqrt(5) - sqrt(3))/(sqrt(5) + sqrt(3))`

Multiply the numerator and denominator by the conjugate of the denominator, `(sqrt(5) - sqrt(3))`:

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

= `(sqrt(5) - sqrt(3))^2/((sqrt(5))^2 - (sqrt(3))^2`

Simplify the denominator (same as before):

`(sqrt(5))^2 - (sqrt(3))^2`

= 5 – 3

= 2

Now expand the numerator:

`(sqrt(5) - sqrt(3))^2`

= `(sqrt(5))^2 - 2sqrt(5)sqrt(3) + (sqrt(3))^2`

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

= `8 - 2sqrt(15)`

So the second term becomes:

`(8 - 2sqrt(15))/2 = 4 - sqrt(15)`

Step 2: Subtract the two terms

Now we subtract the two simplified terms:

`(4 + sqrt(15)) - (4 - sqrt(15))`

Distribute the negative sign:

`4 + sqrt(15) - 4 + sqrt(15) = 2sqrt(15)`