# SAT Math : How to simplify square roots

## Example Questions

### Example Question #1 : Simplifying 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 #2 : Simplifying Square Roots

Simplify .

Explanation:

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

Therefore, .

### Example Question #3 : Simplifying 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 #4 : 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 #5 : Simplifying Square Roots

Which of the following is the most simplified form of:

Explanation:

First find all of the prime factors of

So

### Example Question #6 : 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 #41 : Basic Squaring / Square Roots

Simplify:

Explanation:

Write out the common square factors of the number inside the square root.

Continue to find the common factors for 60.

Since there are no square factors for , the answer is in its simplified form.  It might not have been easy to see that 16 was a common factor of 240.

### Example Question #42 : Basic Squaring / Square Roots

Simplify:

Explanation:

To simplify, we want to find some factors of  where at least one of the factors is a perfect square.

In this case,  and  are factors of , and  is a perfect square.

We can simplify by saying:

We could also recognize that two factors of  are  and . We could approach this way by saying:

But we wouldn't stop there. That's because  can be further factored:

### Example Question #51 : Basic Squaring / Square Roots

Simplify:

Explanation:

To simplify, we want to find factors of  where at least one is a perfect square. With this in mind, we find that:

### Example Question #52 : Basic Squaring / Square Roots

. (Only positive integers)

None of the Above

Explanation:

Step 1: We need to simplify all the roots:

(I am not changing this one, it's already simplified)

Step 2: Rewrite the problem with the simplified parts we found in step 1

Step 3: Combine Like terms:

Numbers:

Roots:

Step 4: Write the final answer. It does not matter how you write it.