Question

Find the value of a and b in the following:

`(3 + sqrt(7))/(3 - sqrt(7)) = a + b sqrt(7)`

Answer 1

We are given:

`(3 + sqrt(7))/(3 - sqrt(7)) = a + b sqrt(7)`

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

`(3 + sqrt(7))/(3 - sqrt(7)) xx (3 + sqrt(7))/(3 + sqrt(7))`

= `(3 + sqrt(7))^2/(3^2 - (sqrt(7))^2`

Step 2: Simplify numerator:

`(3 + sqrt(7))^2`

= `9 + 6sqrt(7) + 7`

= `16 + 6sqrt(7)`

Step 3: Simplify denominator:

9 – 7 = 2

Step 4: Final expression:

`(16 + 6sqrt(7))/2 = 8 + 3sqrt(7)`

Hence, comparing with `a + bsqrt(7)`, we get:

a = 8, b = 3