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© Copyright 1999, Jim Loy
In algebra (and other branches of mathematics) we prefer not to have radicals (square roots including i the square root of -1, cube roots, etc.) in the denominator of a fraction. Sometimes we must go to great lengths do remove these radicals. The process is called rationalizing the denominator. Here you will see 21/2 as the square root of 2. And I am using x for multiplication:
1/21/2=1/21/2 x 21/2/21/2=21/2/2
1/(21/2+1)=1/(21/2+1) x (21/2-1)/(21/2-1)=21/2-1
You can probably see that these final expressions are simpler and easier to handle than the original expressions. You can see why you don't want to divide one by the square root of 2. It is not easy. Dividing the square root of 2 by 2 is much easier.
This is similar to the one that I received email about:
1/(21/3+1)=1/(21/3+1) x (22/3-21/3+1)/(22/3-21/3+1)=(22/3-21/3+1)/3
When asked about this, I couldn't remember how it was done. After being told the answer, it came back to me. What I did here was ask, "What times 21/3+1 gives us an integer?"
(21/3+1) x (22/3+a21/3+b)=c
Solving for a, b, and c, we get a=-1, b=1, and c=3. That gives us 22/3-21/3+1. We multiply both the numerator and denominator by this, and we end up with the difficult stuff in the numerator, and 3 in the denominator.