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Geometry/Rationalizing the Denominator

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Hi Apsara,

Rationalizing the denominator means getting rid of the root on the denominator (thereby making the denominator a rational number).  In principle, there are two types of problems you can encounter.  I will refer to these as "simple radicals" and "mixed radicals".  You may have seen them under different names.  The idea behind both is similar: multiply by something over itself, then multiply across the bottom to remove the radical.

Suppose we wish to rationalize 3/√15, a simple radical.  We are always allowed to multiply by 1, and because anything divided by itself equals 1 (provided the denominator is non-zero), our task is to find something convenient which will cause the radical denominator to be "multiplied out".  With a simple radical, we can always choose the radical divided by itself.  So in our example:

3/√15*(√15/√15)= (3√15)/(√15√15)

Now we multiply out the denominator.

(3√15)/(√15√15) = (3√15)/(15)

Finally, if any numbers can be cancelled or otherwise simplified, do so.  

(3√15)/(15) = √15/5

Thus the rationalized form of 3/√15 is √15/5.


The case of the mixed radical is slightly more complicated, but is really the same thing.  For our multiplication by something over itself, we can always choose the denominator with the sign between the terms flipped.  This is sometimes referred to as the "conjugate".  Let's look at an example.  The denominator is (√3+1), so we will multiply by (√3-1)/(√3-1).  We would select this regardless of the numerator.

(√3-1)/(√3+1) * (√3-1)/(√3-1) = (√3-1)(√3-1)/(√3+1)(√3-1)

Let's split this up and simplify the numerator and denominator separately.

NUMERATOR: (√3-1)(√3-1) = √3√3-√3-√3+1 = 3-2√3+1 = 4-2√3
DENOMINATOR: (√3+1)(√3-1) = √3√3+√3-√3-1 = 3+0-1 = 2

Notice that on the denominator, the cross-terms add up to zero.  It is specifically for this reason that we decided to multiply by the conjugate over itself.

So we now place the numerator over the denominator, and simplify where possible:

(4-2√3)/2 = [2(2-√3)]/2 = 2-√3

And thus (perhaps surprisingly), we have that (√3-1)/(√3+1) = 2-√3.  We have gotten rid of the rational denominator (the denominator is now 1), so we are done.

Many students struggle with this, so be sure to go over it a few times to make sure you understand it, then practice a lot.


Best regards,
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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