Question

If `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`, find y².

Answer 1

Given: `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`

We need to find y².

Step-wise calculation:

1. Rationalise the denominator of y:

`y = (3 - sqrt(2))/(3 + sqrt(2)) xx (3 - sqrt(2))/(3 - sqrt(2))`

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

= `(3 - sqrt(2))^2/(9 - 2)`

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

2. Expand the numerator:

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

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

= `9 + 2 - 6sqrt(2)`

= `11 - 6sqrt(2)`

3. So, `y = (11 - 6sqrt(2))/7`.

4. Now, find y²: 

`y^2 = ((11 - 6sqrt(2))/7)^2`

`y^2 = (11 - 6sqrt(2))^2/7^2`

5. Expand the numerator:

`(11 - 6sqrt(2))^2`

= `11^2 + (6sqrt(2))^2 - 2 xx 11 xx 6sqrt(2)`

= `121 + 72 - 132sqrt(2)`

= `193 - 132sqrt(2)`

6. Thus, `y^2 = (193 - 132sqrt(2))/49`.