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

## Example Questions

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

Which of the following is equivalent to:

?

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

Simplify:

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 #31 : Arithmetic

Simplify the following:

It cannot be simplified any further

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

Simplify the following:

The expression cannot be simplified any further.

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

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 #1 : Factoring Common Factors Of Squares And Square Roots

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