What does it mean when we say "rationalize the denominator"?
It means to have the denominator as a rational number.
Back in the days before electronic calculators became the norm, (I got my first one in 1969),
a question such as 5/(4 - √3) would require a long division of 5 by 2.26795
First of all I had to round off my divisor, and secondly this was a tedious division.
so by multiplying 5/(4 - √3) by (4 + √3)/(4 + √3), which is called its "conjugate"
we would get
5/(4 - √3) * (4 + √3)/(4 + √3)
= (20 + 5√3)/16-9)
= (20 + 5√3)/7
= appr 28.66025/7 , which is a much easier long division.
This justification for rationalizing the denominator is no longer valid, since your
calculator doesn't care if you divide by 7 or by 2.26795
but it is still much valuable in algebra, Calculus and other higher levels of math.