Question

Find the value of a and b in the following:

`(5 + sqrt(5))/(9 - 4sqrt(5)) = a + bsqrt(5)`

Answer 1

We are given the equation:

`(5 + sqrt(5))/(9 - 4sqrt(5)) = a + bsqrt(5)`

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

The conjugate of `9 - 4sqrt(5)` is `9 + 4sqrt(5)`.

Multiply the numerator and denominator by `9 + 4sqrt(5)`:

`(5 + sqrt(5))/(9 - 4sqrt(5)) xx (9 + 4sqrt(5))/(9 + 4sqrt(5))`

= `((5 + sqrt(5))(9 + 4sqrt(5)))/((9 - 4sqrt(5))(9 + 4sqrt(5))`

Step 2: Simplify the denominator using the identity `(a - b)(a + b) = a^2 - b^2`:

`(9 - 4sqrt(5))(9 + 4sqrt(5))`

= `9^2 - (4sqrt(5))^2`

= 81 – 16 × 5

= 81 – 80

= 1

Step 3: Expand the numerator:

Now, expand the numerator `(5 + sqrt(5))(9 + 4sqrt(5))` using distributive property (FOIL):

1. 5 × 9

= 45

2. `5 xx 4sqrt(5)`

= `20sqrt(5)`

3. `sqrt(5) xx 9`

= `9sqrt(5)`

4. `sqrt(5) xx 4sqrt(5)`

= 4 × 5 

= 20

So, the numerator becomes:

`45 + 20sqrt(5) + 9sqrt(5) + 20 = 65 + 29sqrt(5)`

Step 4: Simplify:

Since the denominator is 1, the expression simplifies to `65 + 29sqrt(5)`

a = 65, b = 29