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Squares and Square Roots

Squares and Square Roots

Addition and Subtraction

To add or subtract radical expressions, simplify each radical term and then combine like terms. A simplified radical term consists of a coefficient and a radical, under which there is a radicand. Can you believe you understand now what all these crazy words mean? It's like you can speak a secret language.

Sample Problem

In the term , the coefficient is 5 and the radicand is 7.

Sample Problem

In the term , we need to rationalize the denominator to find .

To put this in the form we need, it could be rewritten as

so the coefficient is and the radicand is 35. We still haven't gotten rid of the fraction line, but at least it isn't combined with the square root symbol any longer. That many weird symbols consorting together makes us nervous. It feels like they're up to something.

Two radical terms are considered like terms if they have the same radicand. This makes them "term twins." You'll be able to tell, because they're always finishing each other's sentences.

Be careful: It's important to simplify radical terms before combining like terms. Sometimes two terms can be rewritten to be like terms, but we can't see it until we simplify. It's the same way you can't switch one kid out for another without their parents noticing until you've first made sure that they're twins. Otherwise, it's called kidnapping.

Radical expressions may contain variables either outside or inside the radicands. After simplifying, we can combine like terms in the same way we did when only numbers were involved. Except now it's more fun, because we can use variables!

...we'll keep telling ourselves that until it feels true.

Now that we know how to figure out which terms can be combined, we'll combine some. How about that.

This is similar to adding or subtracting variables. In the same way that 3x + 4x = 7x, so does .

To add or subtract like radical terms, we add or subtract the coefficients. We don't do anything to the radicands, which is why we made sure they were the same in the first place.

Sample Problems

Do the arithmetic and simplify your answer.

1. .

We keep the radicand the same, and add the coefficients 3 and 8 to find .


First, simplify each radical. Then, the problem can be rewritten as

or ,

which equals .


We can simplify the first radical term and rewrite the problem as


We can't combine these terms since the radicands are not the same, so that's our final answer.

We can also do this with expressions that have variables. Variables and numbers have sort of an "Anything you can do, I can do better" relationship, or at least an "Anything you can do, I can do equally" one.

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