SOLUTION: Rationalize the denominator and simplify. sqrt(6)/4sqrt(3) + sqrt(2) I am having a real hard time figuring the square root problems. I will try: sqrt(6)/4sqrt(3) + sqrt

Algebra ->  Square-cubic-other-roots -> SOLUTION: Rationalize the denominator and simplify. sqrt(6)/4sqrt(3) + sqrt(2) I am having a real hard time figuring the square root problems. I will try: sqrt(6)/4sqrt(3) + sqrt      Log On


   

Question 241032: Rationalize the denominator and simplify.
sqrt(6)/4sqrt(3) + sqrt(2)
I am having a real hard time figuring the square root problems.
I will try: sqrt(6)/4sqrt(3) + sqrt(2) =
sqrt(6)/sqrt(12) + sqrt(8) = sqrt(6)/sqrt(20) =
sqrt(3)/sqrt(10) answer
I am not sure I am doing this problem correctly?
Any help would be really appreciated.

Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
This is a problem called "Rationalizing a BINOMIAL Denominator!" To rationalize a denominator means to "get rid of the radicals in the denominator." In the case of a binomial denominator, it takes a special trick to do what needs to be done here. You have to multiply the numerator and denominator by what is called the "CONJUGATE" of the denominator. That is, the SAME as the denominator, but with the OPPOSITE sign in the middle!! We do this for one reason--because it WORKS!! In this case, you have to multiply the numerator and denominator by 4%2Asqrt%283%29-sqrt%282%29+. It looks like this:

sqrt%286%29%2F%284sqrt%283%29+%2B+sqrt%282%29%29
sqrt%286%29%2F%284sqrt%283%29+%2B+sqrt%282%29%29*%284sqrt%283%29-sqrt%282%29%29%2F%284sqrt%283%29-sqrt%282%29%29+

Now, multiply the numerator and denominator out. Notice that the denominator is left with NO RADICALS!


In the denominator, notice that the middle terms subtract out, and the other denominator terms are PERFECT SQUARES! Simplify the numberator!
%284%2Asqrt%289%29%2Asqrt%282%29+-sqrt%284%29%2Asqrt%283%29%29%2F%2816%2A3-2%29
%284%2A3%2Asqrt%282%29+-2%2Asqrt%283%29%29%2F%2848-2%29+
%2812sqrt%282%29+-+2%2Asqrt%283%29%29%2F46

Factor the numerator so you can reduce the fraction:
%282%286sqrt%282%29-sqrt%283%29%29%29%2F46

%286sqrt%282%29-sqrt%283%29%29%2F23

FINAL ANSWER

I have a LOT of resources on my own website on the topic of SQUARE ROOTS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "MATH IN LIVING COLOR." For problems like this one, please see "Intermediate Algebra" and look in "Chapter 3" for "Section 3.04 Rationalizing Denominators." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Intermediate Algebra", and look for "Radicals and Fractional Exponents".

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus