Question

If `1/(2 + sqrt(3)) = a + bsqrt(3)` then a and b respectively are ______.

Options

• 2, 3

• 2, 1

• 2, –1

• 2, –3

Answer 1

If `1/(2 + sqrt(3)) = a + bsqrt(3)` then a and b respectively are 2, –1.

Explanation:

We need to simplify:

`1/(2 + sqrt(3))`

Step 1: Rationalize the denominator

Multiply numerator and denominator by the conjugate `2 - sqrt(3)`:

`1/(2 + sqrt(3)) xx (2 - sqrt(3))/(2 - sqrt(3))`

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

Step 2: Simplify denominator

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

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

= 4 – 3

= 1

So, `1/(2 + sqrt(3)) = 2 - sqrt(3)`

Step 3: Compare with form `a + bsqrt(3)`

Here, `2 - sqrt(3) = a + bsqrt(3)`

Thus, a = 2, b = –1.