GRE Math : How to find the common factor of square roots

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #1 : How To Find The Common Factor Of Square Roots

Which of the following is equivalent to:

?

Possible Answers:

Correct answer:

Explanation:

To begin with, factor out the contents of the radicals.  This will make answering much easier:

They both have a common factor .  This means that you could rewrite your equation like this:

This is the same as:

These have a common .  Therefore, factor that out:

Example Question #2 : How To Find The Common Factor Of Square Roots

Simplify:

Possible Answers:

Correct answer:

Explanation:

These three roots all have a  in common; therefore, you can rewrite them:

Now, this could be rewritten:

Now, note that 

Therefore, you can simplify again:

Now, that looks messy! Still, if you look carefully, you see that all of your factors have ; therefore, factor that out:

This is the same as:

Example Question #1 : How To Find The Common Factor Of Square Roots

Simplify the following:

Possible Answers:

It cannot be simplified any further

Correct answer:

Explanation:

Begin by factoring each of the roots to see what can be taken out of each:

These can be rewritten as:

Notice that each of these has a common factor of .  Thus, we know that we can rewrite it as:

Example Question #2 : How To Find The Common Factor Of Square Roots

Simplify the following:

Possible Answers:

The expression cannot be simplified any further.

Correct answer:

Explanation:

Clearly, all three of these roots have a common factor  inside of their radicals. We can start here with our simplification. Therefore, rewrite the radicals like this:

We can simplify this a bit further:

From this, we can factor out the common :

Example Question #3 : How To Find The Common Factor Of Square Roots

Possible Answers:

Correct answer:

Explanation:

To attempt this problem, attempt to simplify the roots of the numerator and denominator:

Notice how both numerator and denominator have a perfect square:

The  term can be eliminated from the numerator and denominator, leaving

Example Question #4 : How To Find The Common Factor Of Square Roots

Possible Answers:

Correct answer:

Explanation:

For this problem, begin by simplifying the roots. As it stands, numerator and denominator have a common factor of  in the radical:

And as it stands, this  is multiplied by a perfect square in the numerator and denominator:

The  term can be eliminated from the top and bottom, leaving

Example Question #5 : How To Find The Common Factor Of Square Roots

Possible Answers:

Correct answer:

Explanation:

To solve this problem, try simplifying the roots by factoring terms; it may be noticeable from observation that both numerator and denominator have a factor of  in the radical:

We can see that the denominator has a perfect square; now try factoring the  in the numerator:

We can see that there's a perfect square in the numerator:

Since there is a  in the radical in both the numerator and denominator, we can eliminate it, leaving

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