Question

Find the value of a and b in the following:

`sqrt(5)/(3 - sqrt(5)) = a + bsqrt(5)`

Answer 1

We are given:

`sqrt(5)/(3 - sqrt(5)) = a + bsqrt(5)`

Step 1: Multiply numerator and denominator by the conjugate of the denominator:

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

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

Step 2: Simplify numerator:

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

Step 3: Simplify denominator:

9 – 5 = 4

Step 4: Final expression:

`(3sqrt(5) + 5)/4 = 5/4 + 3/4 sqrt(5)`

`a = 5/4, b = 3/4`