ACT Math : How to simplify square roots

Study concepts, example questions & explanations for ACT Math

varsity tutors app store varsity tutors android store

Example Questions

← Previous 1

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

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

Possible Answers:

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

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

a^{2}bc\sqrt{bc}

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

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

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 #3 : How To Simplify Square Roots

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 #68 : Arithmetic

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

Possible Answers:

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

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

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

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

\frac{3x^{2} - \sqrt{6}}{9x^{2} + 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 #2 : 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 #71 : Arithmetic

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 #3 : Simplifying Square Roots

Which of the following is equal to ?

Possible Answers:

Correct answer:

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 #5 : Simplifying Square Roots

Which of the following is equivalent to ?

Possible Answers:

Correct answer:

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 #6 : How To Simplify Square Roots

What is the simplified (reduced) form of ?

Possible Answers:

It cannot be simplified further.

Correct answer:

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 #6 : Simplifying Square Roots

Which of the following is equal to ?

Possible Answers:

Correct answer:

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.

← Previous 1
Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: