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

## Example Questions

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### Example Question #1 : Basic Squaring / Square Roots

Explanation:

To solve the equation , we can first factor the numbers under the square roots.

When a factor appears twice, we can take it out of the square root.

Now the numbers can be added directly because the expressions under the square roots match.

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

Solve for .

Explanation:

First, we can simplify the radicals by factoring.

Now, we can factor out the .

Now divide and simplify.

### 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 #2 : 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 #3 : Basic Squaring / Square Roots

Simplify:

Explanation:

Begin by factoring out the relevant squared data:

is the same as

This can be simplified to:

Since your various factors contain square roots of , you can simplify:

Technically, you can factor out a :

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

Solve for :

Explanation:

Begin by breaking apart the square roots on the left side of the equation:

This can be rewritten:

You can combine like terms on the left side:

Solve by dividing both sides by :

This simplifies to:

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

Solve for

Explanation:

To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

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Pull  out of each term on the left:

--->

Next, factor out  from the left-hand side:

--->

Lastly, isolate :

--->

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

Solve for

Explanation:

Solving this one is tricky. At first glance, we have no common perfect square to work with. But since each term can produce the quantity , let's start there:

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Simplify the first term:

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Divide all terms by  to simplify,

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Next, factor out  from the left-hand side:

--->

Isolate  by dividing by  and simplifying:

--->

Last, simplify the denominator:

---->

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

Solve for :

Explanation:

Right away, we notice that  is a prime radical, so no simplification is possible. Note, however, that both other radicals are divisible by .

Our first step then becomes simplifying the equation by dividing everything by :

--->

Next, factor out  from the left-hand side:

--->

Lastly, isolate :

--->

### Example Question #8 : Basic Squaring / Square Roots

Solve for

Explanation:

Once again, there are no common perfect squares under the radical, but with some simplification, the equation can still be solved for :

--->

Simplify:

--->

Factor out  from the left-hand side:

--->

Lastly, isolate :

--->

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