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Square Roots: Introduction & Simplification (page 1 of 4)

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

You already know about squaring. For instance, 2² = 4, 3² = 9, etc. The backwards of squaring is square-rooting. The symbol for square-rooting is "", the "radical" symbol. It is used like this:

Now, you can take any counting number and square it, and end up with a nice neat number. But it doesn't work going backwards. Think about . There is no nice neat number that squares to 3. Then can be handled in either of two ways. If you are doing a word problem, for instance, and are trying to find something like, say, speed, then grab your calculator and find the decimal approximation of :

...and round to an appropriate number of decimal places, like "1.7 ft/sec". On the other hand, you may be solving a plain old math problem (with no practical application), in which case you will almost certainly want the "exact" answer, so you'll just leave the answer as "".

When you add x's, you do it in the manner of 2x + 3x = 5x. We do the same with radicals:

• Simplify:  .

Adding and subtracting radicals is similar in ways to adding and subtracting polynomial terms. Just as you can not combine 2x and 3y (because they are not "like terms"), so also you can not combine . You can not combine 2 and 3x (because they are not "like terms"); likewise, you can not combine .   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

You will generally have to "simplify" square root expressions. Some are easy to do:

• Simplify:  .

Don't assume that expressions with unlike radicals cannot be simplified, however. It is possible that, after simplifying the radicals, the expression can indeed be simplified. For instance:

• Simplify:  .

Here is an important property of square roots:

• Write as the product of two radicals:  .

How was I able to rearrange the original radical like that? Because square roots are flexible with multiplication. You can factor the insides of a square root, and then split the square root according to the factors. Sometimes it helps to manipulate the multiplication in the other direction:

• Simplify by writing as one radical:  



                   = 30

Here is an example of how you can use this multiplication property to simplify radical expressions:

• Simplify: .

In general, how do you figure out what can "come out" of a square root? Factor the innards, and any factor that occurs in pairs can come out. For example:

• Simplify: .



 

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