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ACT Math : Simplifying Square Roots

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Example Questions

Example Question #1 : How To Simplify Square Roots

Which of the following is equal to $\sqrt{75}$ ?

Possible Answers:

$3\sqrt{5}$

$5\sqrt{3}$

$7.5\sqrt{10}$

Correct answer:

$5\sqrt{3}$

Explanation:

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

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Example Question #1 : How To Simplify Square Roots

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

Possible Answers:

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

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

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

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

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

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

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Example Question #1 : Simplifying Square Roots

What is $\sqrt{50}$ ?

Possible Answers:

$2\sqrt{5}$

$5\sqrt{2}$

$10\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves $\sqrt{2}$ which can not be simplified further.

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Example Question #2 : How To Simplify Square Roots

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

Possible Answers:

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

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

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

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

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

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:

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

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

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

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

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Example Question #3 : How To Simplify Square Roots

Which of the following is the most simplified form of:

$\sqrt{468}$

Possible Answers:

$\sqrt{468}$

$6\sqrt{13}$

$17\sqrt{2}$

$2\sqrt{117}$

$4\sqrt{29}$

Correct answer:

$6\sqrt{13}$

Explanation:

First find all of the prime factors of 468

$468=6\ast78=6\ast6\ast13=2\ast3\ast2\ast3\ast13$

So $\sqrt{468}=\sqrt{2\ast2\ast3\ast3\ast13}=2\ast3\sqrt{13}=6\sqrt{13}$

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Example Question #1 : How To Simplify Square Roots

What is $\sqrt{432}$ equal to?

Possible Answers:

$6\sqrt{3}$

$12\sqrt{3}$

$144\sqrt{3}$

$6\sqrt{4}$

$12\sqrt{12}$

Correct answer:

$12\sqrt{3}$

Explanation:

$\sqrt{432}$

1. We know that 432=(144)(3) , which we can separate under the square root:

$\sqrt{144\cdot 3}$

2. 144 can be taken out since it is a perfect square: $12\cdot12=144$ . This leaves us with:

$12\sqrt{3}$

This cannot be simplified any further.

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Example Question #2 : Simplifying Square Roots

Which of the following is equal to $\small \sqrt{420}$ ?

Possible Answers:

$\small 28\sqrt{5}$

$\small 2\sqrt{105}$

$\small \small 3\sqrt{70}$

$\small 4\sqrt{35}$

$\small 6\sqrt{35}$

Correct answer:

$\small 2\sqrt{105}$

Explanation:

When simplifying square roots, begin by prime factoring the number in question. For $\small 420$ , this is:

$\small 420 = 2*2*3*5*7$

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

$\small \small \sqrt{420}=2\sqrt{105}$

Another way to think of this is to rewrite $\small \small \sqrt{420}$ as $\small \sqrt{4}*\sqrt{105}$ . This can be simplified in the same manner.

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Example Question #4 : Simplifying Square Roots

Which of the following is equivalent to $\small \sqrt{700700}$ ?

Possible Answers:

$\small 121\sqrt{7}$

$\small 90\sqrt{11}$

$\small 770\sqrt{13}$

$\small 100\sqrt{77}$

$\small 70\sqrt{143}$

Correct answer:

$\small 70\sqrt{143}$

Explanation:

When simplifying square roots, begin by prime factoring the number in question. This is a bit harder for $\small 700700$ . Start by dividing out $\small 100$ :

$\small 700700 = 7007 * 100$

Now, $\small 7007$ is divisible by , so:

$\small \small 700700 =1001 * 7 * 2*2*5*5$

$\small 1001$ is a little bit harder, but it is also divisible by $\small 7$ , so:

$\small 700700 =143*7 * 7 * 2*2*5*5$

With some careful testing, you will see that $\small 143 = 11 * 13$

Thus, we can say:

$\small \small 700700 = 2*2*5*5*7 * 7*11*13$

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

$\small \small \sqrt{700700} = 2*5*7*\sqrt{143}=70\sqrt{143}$

Another way to think of this is to rewrite $\small \small \small \sqrt{700700}$ as $\small \sqrt{4} * \sqrt{25} * \sqrt{49} * \sqrt{143}$ . This can be simplified in the same manner.

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Example Question #2 : Simplifying Square Roots

What is the simplified (reduced) form of $\sqrt{96}$ ?

Possible Answers:

$8\sqrt{3}$

It cannot be simplified further.

$2\sqrt{24}$

$4\sqrt{6}$

$2\sqrt{48}$

Correct answer:

$4\sqrt{6}$

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
$\\ \sqrt{96}=\sqrt{2\cdot 2\cdot 2\cdot2\cdot2\cdot 3}\\ \\=2\cdot 2\sqrt{2\cdot 3}\\ \\=4\sqrt{6}$

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Example Question #51 : Basic Squaring / Square Roots

Which of the following is equal to $\small \sqrt{1260}$ ?

Possible Answers:

$\small 6\sqrt{35}$

$\small 60\sqrt{7}$

$\small 26\sqrt{35}$

$\small 10\sqrt{7}$

$\small 30\sqrt{3}$

Correct answer:

$\small 6\sqrt{35}$

Explanation:

When simplifying square roots, begin by prime factoring the number in question. For $\small 1260$ , this is:

$\small 1260=2*2*3*3*5*7$

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

$\small \sqrt{1260}=2*3*\sqrt{35}=6\sqrt{35}$

Another way to think of this is to rewrite $\small \sqrt{1260}$ as $\small \sqrt{4}*\sqrt{9}*\sqrt{35}$ . This can be simplified in the same manner.

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ACT Math : Simplifying Square Roots

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Simplify Square Roots

Example Question #1 : How To Simplify Square Roots

Example Question #1 : Simplifying Square Roots

Example Question #2 : How To Simplify Square Roots

Example Question #3 : How To Simplify Square Roots

Example Question #1 : How To Simplify Square Roots

Example Question #2 : Simplifying Square Roots

Example Question #4 : Simplifying Square Roots

Example Question #2 : Simplifying Square Roots

Example Question #51 : Basic Squaring / Square Roots

All ACT Math Resources

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