Question

How do you rationalize the denominator and simplify `(4+sqrt3)/(5+sqrt2)`?

Answer 1

To rationalize the denominator we must get rid of any radicals in the denominator.

Explanation:

We can rationalize the denominator of a binomial by multiplying the numerator and the denominator by the conjugate of the denominator.

Definition: the conjugate is what forms a difference of squares. Ex the conjugate of `(a + b)` is `(a - b)`, since `(a + b)(a - b) = a^2 - b^2`. Essentially, to find the conjugate all you have to do is simply to switch the sign `+ or -` sign between the two terms in the binomial.

Thus, the conjugate of `5 + sqrt(2)` is `5 - sqrt(2)`.

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

Don't forget that you cannot multiply non radicals with radicals. E.g `3 xx sqrt(8) != sqrt(24)` but is simply `3sqrt(8)`.

`-> (20 + 5sqrt(3) - 4sqrt(2) - sqrt(6))/(25 + 2sqrt(5) - 2sqrt(5) - sqrt(4)`

= `(20 + 5sqrt(3) - 4sqrt(2) - sqrt(6))/23`

Practice exercises:

Rationalize the denominator of `(3sqrt(5) - 2sqrt(7))/(2sqrt(3) - 4sqrt(2))`

Good luck!