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

1 Answer
Noah G
Mar 13, 2016

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!

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