Question

How do you simplify `4/(sqrtx-sqrty)`?

Answer 1

Rationalize the denominator.

Explanation:

You do this by multiplying the expression by the conjugate of the denominator. The conjugate is found by switching the sign between the terms in the denominator. Hence, in the expression `1/(sqrt(3) + sqrt(5))`, the conjugate of the denominator is `sqrt(3) - sqrt(5)`. We use this technique to get rid of radicals in denominators, because generally, a rational denominator is much more approved mathematically than an irrational denominator. This process of ridding the denominator of irrational numbers is logically named rationalizing the denominator.

Back to our problem at hand. The conjugate of the denominator is `sqrt(x) + sqrt(y)`.

`=4/(sqrt(x) - sqrt(y)) xx (sqrt(x) + sqrt(y))/(sqrt(x) + sqrt(y))`

`= (4sqrt(x) + 4sqrt(y))/(sqrt(x^2) - sqrt(y^2)`

`=(4sqrt(x) + 4sqrt(y))/(x - y)`

`=(4(sqrt(x) + sqrt(y)))/(x - y)`

Practice exercises:

1. Rationalize the denominators of each expression. Make sure to simplify each expression at the end when necessary.

a) `sqrt(3)/(sqrt(2) + sqrt(7)`

b) `(sqrt(5) + 2sqrt(2))/(sqrt(11) - sqrt(15)`

c) `(4x)/(sqrt(2x) + sqrt(3y)`

Challenge Problem:

Rationalize the denominator:

`(3)/(root(3)(7) - root(3)(6))`

Hopefully this helps, and good luck!