How do we rationalize a denominator?

Question

Zoe Adams · Accepted Answer

Multiply the whole fraction by the denominator over itself.

If you have a fraction where the denominator (bottom number) is irrational, such as a square root, then multiply the whole fraction by the denominator divided by itself. Take the example #5/sqrt2# which has a rational denominator. Multiply the entire fraction by the denom. divided by itself, like so #5/sqrt 2 * sqrt2/sqrt2 = (5sqrt2)/2# More examples are #3/sqrt7 * sqrt7/sqrt7 = (3sqrt7)/7# #5/sqrt18 = 5/(3sqrt2)# #5/(3sqrt2) * sqrt2/sqrt2 = (5sqrt2)/6# In this last example I turned #sqrt18# into #3sqrt2# by laws of surds, and then, because #3# is already rational, I only multiplied by #sqrt2/sqrt2#, not #(3sqrt2)/(3sqrt2)#.

Riley Anderson · Answer

undefined To rationalize a denominator, you multiply both the numerator and the denominator of the fraction by the conjugate of the denominator. The conjugate of a binomial is obtained by changing the sign between the terms. This process eliminates radicals (square roots or cube roots, etc.) from the denominator. The result is a fraction with a rational (non-radical) denominator.