SOLUTION: what is 6 divided by the square root of 2 minus the square root of 3?

Question 712418: what is 6 divided by the square root of 2 minus the square root of 3?

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
:
This is not so much a problem of division but a problem of rationalizing a denominator. The two square roots in the denominator are irrational numbers. Proper form for fractions is to have a rational denominator. So our problem is to find a way to turn the denominator into a rational number.

It is a little harder to do this with two terms in the denominator (compared to one term). The "trick" is to take advantage of the pattern:

One way to look at this pattern is that it shows us how to turn a binomial (an a+b or a-b) and turn it into an expression of perfect squares. This shoes us how to take a binomial with square roots and turn it into an expression of the squares of those square roots. Since our denominator has a "-" between the terms, it will play the role of a-b with the "a" being and the "b" being . To turn it into an expression of perfect squares we need to multiply it by a+b. And of course if we multiply the denominator by something we need to multiply the numerator by the same thing. So we'll multiply the numerator and denominator by :

On top we will simply use the Distributive Property to multiply. On the bottom, the pattern tells us how it will work out. (FOIL can be used but it is slower.)

The squared square roots will simplify:

We now have the rational denominator we wanted. Continuing to simplify...

This is the simplified expression.

P.S. In this problem, the fraction disappeared after we rationalized the denominator. This will not always happen. Often the fraction you have at the end will have square roots in the numerator. For reasons I cannot explain, irrational numbers like square roots are OK in numerators but not OK in denominators.
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