# ACT Math : How to simplify square roots

## Example Questions

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### Example Question #1 : How To Simplify Square Roots

Which of the following is equal to  ?

Explanation:

√75 can be broken down to √25 * √3. Which simplifies to 5√3.

### Example Question #1 : How To Simplify Square Roots

Simplify .

Explanation:

Rewrite what is under the radical in terms of perfect squares:

Therefore, .

### Example Question #41 : Basic Squaring / Square Roots

What is ?

Explanation:

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves  which can not be simplified further.

### Example Question #1 : Simplifying Square Roots

Which of the following is equivalent to ?

Explanation:

Multiply by the conjugate and the use the formula for the difference of two squares:

### Example Question #1 : How To Simplify Square Roots

Which of the following is the most simplified form of:

Explanation:

First find all of the prime factors of

So

### Example Question #2 : Simplifying Square Roots

What is  equal to?

Explanation:

1. We know that , which we can separate under the square root:

2. 144 can be taken out since it is a perfect square: . This leaves us with:

This cannot be simplified any further.

### Example Question #2 : How To Simplify Square Roots

Which of the following is equal to ?

Explanation:

When simplifying square roots, begin by prime factoring the number in question. For , this is:

Now, for each pair of numbers, you can remove that number from the square root. Thus, you can say:

Another way to think of this is to rewrite  as . This can be simplified in the same manner.

### Example Question #1 : Simplifying Square Roots

Which of the following is equivalent to ?

Explanation:

When simplifying square roots, begin by prime factoring the number in question. This is a bit harder for . Start by dividing out :

Now,  is divisible by , so:

is a little bit harder, but it is also divisible by , so:

With some careful testing, you will see that

Thus, we can say:

Now, for each pair of numbers, you can remove that number from the square root. Thus, you can say:

Another way to think of this is to rewrite  as . This can be simplified in the same manner.

### Example Question #3 : Simplifying Square Roots

What is the simplified (reduced) form of ?

It cannot be simplified further.

Explanation:

To simplify a square root, you have to factor the number and look for pairs. Whenever there is a pair of factors (for example two twos), you pull one to the outside.

Thus when you factor 96 you get

### Example Question #71 : Arithmetic

Which of the following is equal to ?