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SAT Math : How to simplify square roots

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Example Questions

Example Question #11 : How To Simplify Square Roots

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

Possible Answers:

$7.5\sqrt{10}$

$3\sqrt{5}$

$5\sqrt{3}$

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 #23 : Arithmetic

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

Possible Answers:

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

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

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

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

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

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 #2 : Simplifying Square Roots

What is $\sqrt{50}$ ?

Possible Answers:

$2\sqrt{5}$

$10\sqrt{2}$

$5\sqrt{2}$

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 #25 : Arithmetic

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

Possible Answers:

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

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

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

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

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

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 : Simplifying Square Roots

Which of the following is the most simplified form of:

$\sqrt{468}$

Possible Answers:

$\sqrt{468}$

$17\sqrt{2}$

$2\sqrt{117}$

$6\sqrt{13}$

$4\sqrt{29}$

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 #11 : How To Simplify Square Roots

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

Possible Answers:

$6\sqrt{3}$

$6\sqrt{4}$

$12\sqrt{12}$

$12\sqrt{3}$

$144\sqrt{3}$

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 #44 : Basic Squaring / Square Roots

Simplify: $\sqrt{240}$

Possible Answers:

$4\sqrt{15}$

$15\sqrt2$

$5\sqrt{13}$

$12\sqrt{4}$

$3\sqrt{27}$

Correct answer:

$4\sqrt{15}$

Explanation:

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

$\sqrt{240}= \sqrt{4\times 60}= 2\sqrt{60}$

Continue to find the common factors for 60.

$2\sqrt{60} = 2\sqrt{4\times 15} = 2(2\sqrt{15}) = 4\sqrt{15}$

Since there are no square factors for $\sqrt{15}$ , 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: $4\sqrt{15}$

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Example Question #11 : How To Simplify Square Roots

Simplify:

$\sqrt{48}$

Possible Answers:

None of the given answers.

$2\sqrt{24}$

$4\sqrt{3}$

$3\sqrt{4}$

Correct answer:

$4\sqrt{3}$

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:

$\sqrt{48} = \sqrt{16\cdot 3} = \sqrt{4^2 \cdot 3} = 4\sqrt{3}$

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

$\sqrt{48}=\sqrt{4 \cdot 12} = \sqrt {2^2 \cdot 12} = 2\sqrt{12}$

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

$2\sqrt{12} = 2\sqrt{4 \cdot 3} = 2\sqrt{2^2 \cdot 3} = 2\cdot 2 \sqrt{3} = 4\sqrt{3}$

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Example Question #12 : How To Simplify Square Roots

Simplify:

$\sqrt{20,000}$

Possible Answers:

$10\sqrt{2}$

$1000\sqrt{2}$

$10\sqrt{20}$

$100\sqrt{2}$

None of the given answers.

Correct answer:

$100\sqrt{2}$

Explanation:

To simplify, we want to find factors of 20,000 where at least one is a perfect square. With this in mind, we find that:

$\sqrt{20,000} = \sqrt{2 \cdot 10,000} = \sqrt{2 \cdot 100^2} = 100\sqrt{2}$

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Example Question #17 : Simplifying Square Roots

Simplify and add:

$\sqrt {64}+\sqrt {27}-\sqrt {81}+\sqrt {3}$ . (Only positive integers)

Possible Answers:

None of the Above

$4\sqrt{3}+1$

$-4\sqrt{3}-1$

$4\sqrt {3}-1$

Correct answer:

$4\sqrt {3}-1$

Explanation:

Step 1: We need to simplify all the roots:

$\sqrt{64}=\sqrt{16}\cdot\sqrt{4}=4\cdot2=8$

$\sqrt {27}=\sqrt {9}\cdot\sqrt {3}=3\cdot\sqrt{3}=3\sqrt{3}$

$\sqrt {81}=\sqrt{9}\cdot\sqrt{9}=3*3=9$

$\sqrt{3}\rightarrow\sqrt{3}$ (I am not changing this one, it's already simplified)

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

$8+3\sqrt{3}-9+\sqrt{3}$

Step 3: Combine Like terms:

Numbers: 8-9=-1

Roots: $3\sqrt{3}+\sqrt{3}=4\sqrt{3}$

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

$4\sqrt{3}-1$

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SAT Math : How to simplify square roots

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #11 : How To Simplify Square Roots

Example Question #23 : Arithmetic

Example Question #2 : Simplifying Square Roots

Example Question #25 : Arithmetic

Example Question #3 : Simplifying Square Roots

Example Question #11 : How To Simplify Square Roots

Example Question #44 : Basic Squaring / Square Roots

Example Question #11 : How To Simplify Square Roots

Example Question #12 : How To Simplify Square Roots

Example Question #17 : Simplifying Square Roots

All SAT Math Resources

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