Question

If a and b are rational numbers, find the value of a and b:

`a + bsqrt(5) = (4 + 2sqrt(5))/(sqrt(5) + 1) - (4 - 2sqrt(5))/(sqrt(5) - 1)`

Answer 1

Given: Find rational numbers a and b such that `a + bsqrt(5) = (4 + 2sqrt(5))/(sqrt(5) + 1) - (4 - 2sqrt(5))/(sqrt(5) - 1)`

Stepwise calculation:

1. Rationalize each denominator separately:

`(4 + 2sqrt(5))/(sqrt(5) + 1) xx (sqrt(5) - 1)/(sqrt(5) - 1)`

= `((4 + 2sqrt(5))(sqrt(5) - 1))/((sqrt(5) + 1)(sqrt(5) - 1))`

= `((4 + 2sqrt(5))(sqrt(5) - 1))/(5 - 1)`

= `((4 + 2sqrt(5))(sqrt(5) - 1))/4`

Expand numerator:

`(4)(sqrt(5)) - 4 + 2sqrt(5) xx sqrt(5) - 2sqrt(5)`

= `4sqrt(5) - 4 + 2 xx 5 - 2sqrt(5)`

= `4sqrt(5) - 4 + 10 - 2sqrt(5)`

= `(4sqrt(5) - 2sqrt(5)) + (-4 + 10)`

= `2sqrt(5) + 6`

So, `(4 + 2sqrt(5))/(sqrt(5) + 1)`

= `(2sqrt(5) + 6)/4`

= `6/4 + (2sqrt(5))/4`

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

2. Similarly for the second term:

`(4 - 2sqrt(5))/(sqrt(5) - 1) xx (sqrt(5) + 1)/(sqrt(5) + 1)`

= `((4 - 2sqrt(5))(sqrt(5) + 1))/((sqrt(5) - 1)(sqrt(5) + 1))`

= `((4 - 2sqrt(5))(sqrt(5) + 1))/(5 - 1)`

= `((4 - 2sqrt(5))(sqrt(5) + 1))/4`

Expand numerator:

`4sqrt(5) + 4 - 2sqrt(5) xx sqrt(5) - 2sqrt(5)`

= `4sqrt(5) + 4 - 2 xx 5 - 2sqrt(5)`

= `4sqrt(5) + 4 - 10 - 2sqrt(5)`

= `(4sqrt(5) - 2sqrt(5)) + (4 - 10)`

= `2sqrt(5) - 6`

So, `(4 - 2sqrt(5))/(sqrt(5) - 1)`

= `(2sqrt(5) - 6)/4`

= `(-6)/4 + (2sqrt(5))/4`

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

3. Now subtract the two:

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

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

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

= 3 + 0

= 3

`a + bsqrt(5) = 3`

Since `3 = 3 + 0 xx sqrt(5)`, we identify a = 3, b = 0.

Hence, the values of a and b are a = 3, b = 0.