Question

Find the value of a and b in the following:

`(3 + sqrt(2))/(10 - 7sqrt(2)) = a + bsqrt(2)`

Answer 1

We are asked to simplify:

`(3 + sqrt(2))/(10 - 7sqrt(2)) = a + bsqrt(2)`

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

The conjugate of `10 - 7sqrt(2)` is `10 + 7sqrt(2)`:

`(3 + sqrt(2))/(10 - 7sqrt(2)) xx (10 + 7sqrt(2))/(10 + 7sqrt(2))`

Step 2: Expand numerator

`(3 + sqrt(2))(10 + 7sqrt(2))`

= `3 xx 10 + 3 xx 7sqrt(2) + sqrt(2) xx 10 + sqrt(2) xx 7sqrt(2)`

= `30 + 21sqrt(2) + 10sqrt(2) + 14`

= `44 + 31sqrt(2)`

Step 3: Expand denominator

`(10 - 7sqrt(2))(10 + 7sqrt(2))`

= `10^2 - (7sqrt(2))^2`

= 100 – 98

= 2

Step 4: Simplify fraction

`(44 + 31sqrt(2))/2 = 22 + 31/2 sqrt(2)`

Step 5: Compare with `a + bsqrt(2)`

`a = 22, b = 31/2`