Question

How do you rationalize the denominator and simplify `6/(3+sqrt3)`?

Answer 1

The answer is `3 - sqrt(3)`.

Explanation:

When rationalizing expressions where the denominators are binomials (two terms), we multiply the entire expression by the conjugate of the denominator.

The conjugate is meant to make a difference of squares when multiplied with its original expression. For example, the conjugate of `a + b` is `a -b`, since `(a + b)(a - b) = a^2 + ab - ab - b^2 = a^2 - b^2`, or a difference of squares.

The conjugate can always be found by switching the middle sign of the denominator. Hence, the conjugate of `3 + sqrt(3)` is `3 - sqrt(3)`.

Let's start the rationalization process.

`=6/(3 + sqrt(3)) xx (3 - sqrt(3))/(3 - sqrt(3))`

`=(18 - 6sqrt(3))/(9 - 3sqrt(3) + 3sqrt(3) - sqrt(9))`

`=(18 - 6sqrt(3))/(9 - 3)`

`=(6(3 - sqrt(3)))/(6)`

`=3 - sqrt(3)`

Hopefully this helps!