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SAT Math : How to simplify square roots

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Example Question #21 : Simplifying Square Roots

Simplify the following:

$2\sqrt{54}$

Possible Answers:

$6\sqrt{6}$

$4\sqrt{3}$

$4\sqrt{2}$

Correct answer:

$6\sqrt{6}$

Explanation:

To solve, you must first break up 54 into its smallest prime factors. Those are:

54=2*27=2*9*3

Since our root has index 2, that means that for every 2 identical factors inside, you can pull 1 out. Thus, we get

$2*3\sqrt{2*3}=6\sqrt6$

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Example Question #22 : Simplifying Square Roots

Simplify $\sqrt{88}$

Possible Answers:

$4\sqrt{11}$

$22\sqrt{2}$

$8\sqrt{11}$

$2\sqrt{22}$

$4\sqrt{22}$

Correct answer:

$2\sqrt{22}$

Explanation:

To simplify a square root, we need to find perfect squares. In this case, it is .

$\sqrt{88}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$

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Example Question #23 : Simplifying Square Roots

Simplify: $\sqrt{332}$

Possible Answers:

$16\sqrt{11}$

$16\sqrt{22}$

$4\sqrt{22}$

$8\sqrt{26}$

Correct answer:

$4\sqrt{22}$

Explanation:

To simplify a square root, we need to find perfect squares. In this case, it is .

$\sqrt{332}=\sqrt{16}*\sqrt{22}=4\sqrt{22}$

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Example Question #24 : Simplifying Square Roots

Simplify: $8\sqrt{8}$

Possible Answers:

$12\sqrt{2}$

$24\sqrt{2}$

$16\sqrt{2}$

$10\sqrt{2}$

Correct answer:

$16\sqrt{2}$

Explanation:

To simplify a square root, we need to find perfect squares. In this case, it is . Since there is a number outside the radical, we ignore that for now and later we multiply the number and square root.

$8\sqrt{8}=8*\sqrt{4}*\sqrt{2}=8*2*\sqrt{2}=16\sqrt{2}$

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Example Question #25 : How To Simplify Square Roots

Simplify: $5\sqrt{98}$

Possible Answers:

$12\sqrt{2}$

$35\sqrt{2}$

$12\sqrt{5}$

$15\sqrt{7}$

$17\sqrt{2}$

Correct answer:

$35\sqrt{2}$

Explanation:

To simplify a square root, we need to find perfect squares. In this case, it is . Since there is a number outside the radical, we ignore that for now and later we multiply the number and square root.

$5\sqrt{98}=5*\sqrt{49}8\sqrt{2}=5*7*\sqrt{2}=35\sqrt{2}$

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Example Question #25 : Simplifying Square Roots

Simplify: $\sqrt{1200}$

Possible Answers:

$16\sqrt{15}$

$20\sqrt{30}$

$400\sqrt{3}$

$40\sqrt{3}$

$20\sqrt{3}$

Correct answer:

$20\sqrt{3}$

Explanation:

To simplify radicals, we need to find a perfect square to factor out. In this case, its 400 .

$\sqrt{1200}=\sqrt{400}*\sqrt{3}=20\sqrt{3}$

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Example Question #26 : Simplifying Square Roots

Simplify: $\sqrt{375}$

Possible Answers:

$15\sqrt{5}$

$25\sqrt{15}$

$3\sqrt{55}$

$5\sqrt{15}$

$5\sqrt{30}$

Correct answer:

$5\sqrt{15}$

Explanation:

To simplify radicals, we need to find a perfect square to factor out. In this case, its .

$\sqrt{375}=\sqrt{25}*\sqrt{15}=5\sqrt{15}$

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Example Question #27 : Simplifying Square Roots

Simplify: $\sqrt{x^5}$

Possible Answers:

$x^2\sqrt{x}$

$x\sqrt{x}$

$x\sqrt{x^3}$

$x^2\sqrt{x^3}$

$2x\sqrt{x}$

Correct answer:

$x^2\sqrt{x}$

Explanation:

To solve this, we know perfect squares are able to simplify easily to the base it is. Let's find all the perfect squares in x^5 .

$\sqrt{x^5}=\sqrt{x^2}*\sqrt{x^2}*\sqrt{x}=x*x\sqrt{x}=x^2\sqrt{x}$

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Example Question #29 : How To Simplify Square Roots

Simplify: $\sqrt{x^{-6}}$

Possible Answers:

$x^{-2}$

x^3

$x^{-3}$

It's impossible because the value is negative.

x^2

Correct answer:

$x^{-3}$

Explanation:

Although the exponent is negative, we know that $x^{-6}=\frac{1}{x^6}$ . Therefore, we have $\sqrt{\frac{1}{x^6}}$ . Let's simplify this by finding perfect squares. $\sqrt{\frac{1}{x^6}}=\sqrt\frac{1}{x^2}*\sqrt\frac{1}{x^2}*\sqrt\frac{1}{x^2}=\frac{1}{x}*\frac{1}{x}*\frac{1}{x}=\frac{1}{x^3}$

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Example Question #28 : Simplifying Square Roots

Simplify: $\sqrt{8^7}$

Possible Answers:

$8^3*2\sqrt{2}$

$128\sqrt{2}$

$8^6\sqrt{2}$

$64\sqrt{2}$

$8^3\sqrt{8}$

Correct answer:

$8^3*2\sqrt{2}$

Explanation:

To solve this, we know perfect squares are able to simplify easily to the base it is. Let's find all the perfect squares in 8^7 .

$\sqrt{8^7}=\sqrt{8^2}*\sqrt{8^2}*\sqrt{8^2}*\sqrt{8}=8^3\sqrt{8}=8^3*2\sqrt{2}$

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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 #21 : Simplifying Square Roots

Example Question #22 : Simplifying Square Roots

Example Question #23 : Simplifying Square Roots

Example Question #24 : Simplifying Square Roots

Example Question #25 : How To Simplify Square Roots

Example Question #25 : Simplifying Square Roots

Example Question #26 : Simplifying Square Roots

Example Question #27 : Simplifying Square Roots

Example Question #29 : How To Simplify Square Roots

Example Question #28 : Simplifying Square Roots

All SAT Math Resources

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