N-RN.2 (Real Number System) - Rationalizing the Denominator

This is the third and final column on Common Core Standard N-RN.2. If you missed the first two parts, Part 1 dealt with evaluating expressions with rational exponents, and Part 2 showed how to simplify using factor trees and how to add and subtract radicals. The last piece of this standard (and since I'm only dealing with part ".2", I could really call it a "substandard" if I wanted to, mockingly) is to "Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)".

Previously, we mentioned that you can multiply two radical numbers by multiplying their radicands. We also factored radical numbers in order to simplify them. Let's talk about division. When you divide, you multiply by the reciprocal; that is, you can create a fraction of the two numbers without relying on your early education "gazintas". (You remember, "2 gazinta 6 three times".) Likewise, when you take the square root of a fraction, you are actually dividing one radical number by another.

So if you wanted the square root of 1/4, you would take the square root of the numerator (radical 1 is 1) over the square root of the denominator (radical 4 is 2). The result would be 1/2.

But suppose we wanted the square root of 1/2? Again, we can split it up into the square root of 1 (which is 1) over the square root of 2.

Here's where we run into a problem because there are rules from fractions. One of them is that there cannot be any radicals in denominator. You have to get rid of them.

We haven't discussed this before, but there's really only one simply way to get rid of a square root sign: square the number. We need to multiply the denominator by radical 2. We are allowed to do this because it's a fraction and we won't change the value of the fraction at all as long as we multiply the numerator by the same amount as the denominator. The fractions (square root of 2 over square root of 2 looks scary to evaluate until you remember that any number, even an irrational, divided by itself is one, with the exception of zero. If you multiply a fraction by 1, it doesn't change its value, even if it looks different. The result is that the radical is gone from the denominator and has moved into the numerator, which is allowed.

One more example. Try it yourself before scrolling down and looking at the image. What is the square root of 4/5?
Take the square root of each number. Rationalize the denominator. What's left in the numerator? What's left in the denominator?

Okay, check your work.

That's it for this standard. Time to move on to part 3, coming soon.