Simplifying / Multiplying Radicals

As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of).

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• Simplify

You are used to putting the numbers first in an algebraic expression, followed by any variables. But for radical expressions, any variables outside the radical should go in front of the radical, as shown above. Always put everything you take out of the radical in front of that radical (if anything is left inside it).

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• Simplify

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Technical point: Your textbook may tell you to "assume all variables are positive" when you simplify. Why? Because the square root of the square of a negative number is not the original number.

For instance, you could start with −2, square it to get +4, and then take the square root of +4 (which is defined to be the positive root) to get +2. You plugged in a negative and ended up with a positive.

But this technicality can cause difficulties if you're working with values of unknown sign; that is, with variables. The |−2| is +2, but what is the sign on | x |? You can't know, because you don't know the sign of x itself — unless they specify that you should "assume all variables are positive", or at least non-negative (which means "positive or zero").

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Multiplying Square Roots

The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.

Simplifying multiplied radicals is pretty simple, being barely different from the simplifications that we've already done. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa.

• Write as the product of two radicals:

Yes, that manipulation was fairly simplistic and wasn't very useful, but it does show how we can manipulate radicals. And using this manipulation in working in the other direction can be quite helpful. For instance:

• Simplify by writing with no more than one radical:

By the way, I could have done the simplification of each radical first, then multiplied, and then does another simplification. The work would be a bit longer, but the result would be the same:

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• Simplify by writing with no more than one radical:

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• Simplify by writing with no more than one radical:

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The process works the same way when variables are included:

• Simplify by writing with no more than one radical: