Question 201697: Rationalize the denominator: 3 / 4-√7
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Rationalizing multiple term denominators, like  , is a little tricky. Multiplying the numerator and denominator by  doesn't work!?
Instead we use the equation  on denominators like these. (You should be familiar with this equation from factoring. If you haven't done factoring yet, then remember this equation.) If you think about this equation, one of the things we can learn which is useful to us here, is that it shows how you can multiply a binomial (2-term expression) like (a+b) or (a-b) and multiply it my something and get nothing but perfect square terms!
Your denominator is of the form (a-b) where  and  . So what do we multiply (a-b) to get nothing but perfect squares? Well, from the equation we can see that it would be (a+b). With your a and b this would be  .
Of course if we want to multiply the denominator by we must also multiply the numerator by it, too:
Use the distributive property on the numerators. In the denominators, we already know that the product is  with and  . (If you don't trust the equation, just use FOIL.)
Simplifying the denominator we get:
And then
And we we have a rational denominator!
Of course, as with all answers that are fractions, we should try to reduce the fraction by canceling common factors. This fraction will reduce. Factor a 3 from the top and bottom:
Cancel the 3's
and we end up with:
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