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

## Example Questions

### Example Question #1 : Factoring And Simplifying Square Roots

Solve for :

Explanation:

Notice how all of the quantities in square roots are divisible by 9

Simplifying, this becomes

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

Solve for :

Explanation:

Note that all of the square root terms share a common factor of 36, which itself is a square of 6:

Factoring  from both terms on the left side of the equation:

### Example Question #2 : Basic Squaring / Square Roots

Solve for :

Explanation:

Note that both  and  have a common factor of  and  is a perfect square:

From here, we can factor  out of both terms on the lefthand side

### Example Question #3 : Basic Squaring / Square Roots

Solve for :

Explanation:

In order to solve for , first note that all of the square root terms on the left side of the equation have a common factor of 9 and 9 is a perfect square:

Simplifying, this becomes:

### Example Question #2 : 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 #4 : Basic Squaring / Square Roots

Solve for :

Explanation:

Examining the terms underneath the radicals, we find that  and  have a common factor of  itself is a perfect square, being the product of  and . Hence, we recognize that the radicals can be re-written in the following manner:

, and .

The equation can then be expressed in terms of these factored radicals as shown:

Factoring the common term  from the lefthand side of this equation yields

Divide both sides by the expression in the parentheses:

Divide both sides by  to yield  by itself on the lefthand side:

Simplify the fraction on the righthand side by dividing the numerator and denominator by :

This is the solution for the unknown variable  that we have been required to find.