SAT Math : How to simplify square roots

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #1 : How To Simplify Square Roots

Which of the following is equal to  ?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #1 : Simplifying Square Roots

Simplify \sqrt{a^{3}b^{4}c^{5}}.

Possible Answers:

a^{2}b^{2}c^{2}\sqrt{bc}

ab^{2}c^{2}\sqrt{ac}

a^{2}b^{2}c\sqrt{ab}

a^{2}bc^{2}\sqrt{ac}

a^{2}bc\sqrt{bc}

Correct answer:

ab^{2}c^{2}\sqrt{ac}

Explanation:

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

x^{2}=x\cdot x

x^{4}=x^{2}\cdot x^{2}

x^{6}=x^{3}\cdot x^{3}

Therefore, \sqrt{a^{3}b^{4}c^{5}}= \sqrt{a^{2}a^{1}b^{4}c^{4}c^{1}}=ab^{2}c^{2}\sqrt{ac}.

Example Question #21 : Arithmetic

What is ?

Possible Answers:

Correct answer:

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 : How To Simplify Square Roots

Which of the following is equivalent to \frac{x + \sqrt{3}}{3x + \sqrt{2}}?

Possible Answers:

\frac{3x^{2} + 3x\sqrt{2} + x\sqrt{3} +\sqrt{6}}{9x^{2} - 2}

\frac{3x^{2} - \sqrt{6}}{9x^{2} + 2}

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

\frac{3x^{2} + \sqrt{6}}{3x - 2}

\frac{4x + \sqrt{5}}{3x + 2}

Correct answer:

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

Explanation:

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

\frac{x + \sqrt{3}}{3x + \sqrt{2}}

\frac{x + \sqrt{3}}{3x + \sqrt{2}}\cdot \frac{3x - \sqrt{2}}{3x - \sqrt{2}}

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{(3x)^{2} - (\sqrt{2})^{2}} 

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

Example Question #1 : How To Simplify Square Roots

Which of the following is the most simplified form of:

 

Possible Answers:

Correct answer:

Explanation:

First find all of the prime factors of 

So 

Example Question #3 : Simplifying Square Roots

What is  equal to?

Possible Answers:

Correct answer:

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 #111 : Arithmetic

Simplify:  

Possible Answers:

Correct answer:

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.

The answer is: 

Example Question #11 : How To Simplify Square Roots

Simplify:

Possible Answers:

None of the given answers. 

Correct answer:

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 #11 : How To Simplify Square Roots

Simplify: 

Possible Answers:

None of the given answers. 

Correct answer:

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 #11 : How To Simplify Square Roots

Simplify and add:

. (Only positive integers)

Possible Answers:

None of the Above

Correct answer:

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.

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