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Rationalizing Radical Denominators (page 4 of 4)

Sections: Square roots, Other roots / Domains, Further simplifying, Rationalizing denominators

For whatever reason, it is generally considered "improper" to leave radicals in the denominator. (This is similar to the fuss regarding improper fractions versus mixed numbers: once you get far enough in math, radicals in the denominator probably won't matter any more, just like, now that you're in algebra, your teacher doesn't get upset about improper fractions any more, like your teacher did back in sixth grade.) The process of getting rid of radicals in the denominator is called "rationalizing the denominator".

• Simplify  .

Now you may think "This looks pretty darn simple already!" But, by "simplify", in this case they mean "get rid of the radical in the denominator". To do this, I will need to turn the into something else. If you think back to when you were dealing with fractions, you would get a common denominator by multiplying non-common denominators by some useful number. For instance, if you needed to turn two-thirds into some kind of nineths, you would multiply by three-over-three:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

You follow this same sort of procedure with rationalizing denominators. If you multiply by another , you'll get "3". So I'll follow the pattern with radicals as we used to use with fractions:

When rationalizing denominators, figure out what you need to have in order to be able to take a square out of the radical (or a cube, if you're dealing with a cube root, etc.), multiply the fraction, top and bottom, by this value, and simplify. Here's another example:

• Simplify  .

This one works just like the previous example, except that I can do some further simplifying at the end:

It is always a good idea to check if your fraction can be simplified. Usually it can't, but be sure to check.

• Simplify  .

This is an instance of where I'll have to simplify, then multiply, and then simplify again. The final answer doesn't look a whole lot "simpler" than what I started with, but it won't have radicals in the denominator, and that's what they're looking for when they say to "simplify".

What about when you have more than one term in the denominator? This is where the "difference of squares" conjugate thing comes in. For example:

• Simplify  .

You know that if you multiply this denominator by its conjugate, the radicals will disappear. So I'll multiply the fraction, top and bottom, by the conjugate:

Since I knew I had created a difference of squares in the denominator (that's the point of using the conjugate), I didn't bother doing the multiplication explicitly, but just converted right to the difference-of-squares format, and then simplified. That's where the "9 – 6" came from.

• Simplify  .

This one works the same way as the problem above:

In the last step, by the way, I took the "minus" from the denominator and put it out front. You can move minuses like that to the front or multiplied through the top; it just isn't considered "standard" to leave them on the denominator.

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