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

1 Answer
Mar 13, 2016

Answer:

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!