Question

Find the value of a and b in the following:

`(6 + sqrt(3))/(7 - 4sqrt(3)) = a + bsqrt(3)`

Answer 1

We are given the equation:

`(6 + sqrt(3))/(7 - 4sqrt(3)) = a + bsqrt(3)`

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

The conjugate of `7 - 4sqrt(3)` is `7 + 4sqrt(3)`.

Multiply the numerator and denominator by `7 + 4sqrt(3)`:

`(6 + sqrt(3))/(7 - 4sqrt(3)) xx (7 + 4sqrt(3))/(7 + 4sqrt(3))`

= `((6 + sqrt(3))(7 + 4sqrt(3)))/((7 - 4sqrt(3))(7 + 4sqrt(3))`

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

`(7 - 4sqrt(3))(7 + 4sqrt(3))`

= `7^2 - (4sqrt(3))^2`

= 49 – 16 × 3

= 49 – 48

= 1

So, the denominator becomes 1.

Step 3: Expand the numerator:

Now, expand the numerator `(6 + sqrt(3))(7 + 4sqrt(3))` using distributive property (FOIL):

1. 6 × 7

= 42

2. `6 xx 4sqrt(3)`

= `24sqrt(3)`

3. `sqrt(3) xx 7`

= `7sqrt(3)`

4. `sqrt(3) xx 4sqrt(3)`

= 4 × 3

= 12

So, the numerator becomes:

`42 + 24sqrt(3) + 7sqrt(3) + 12 = 54 + 31sqrt(3)`

Step 4: Final simplification: 

Since the denominator is 1, the expression simplifies to `54 + 31sqrt(3)`

a = 54, b = 31