Question

Find the value of a and b in the following:

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

Answer 1

We are given the equation:

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

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

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

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

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

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

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

= 4 × 3 – 9

= 12 – 9

= 3

Step 3: Expand the numerator:

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

1. `5sqrt(3) xx 2sqrt(3)`

= 10 × 3

= 30

2. `5sqrt(3) xx 3`

= `15sqrt(3)`

3. `3 xx 2sqrt(3)`

= `6sqrt(3)`

4. 3 × 3

= 9

So the numerator becomes:

`30 + 15sqrt(3) + 6sqrt(3) + 9 = 39 + 21sqrt(3)`

Step 4: Combine and simplify:

Now we can write the expression as:

`(39 + 21sqrt(3))/3`

Divide each term by 3:

= `39/3 + (21sqrt(3))/3`

= `13 + 7sqrt(3)`

a = 13, b = 7