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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #61 : Basic Squaring / Square Roots

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

Possible Answers:

x^3

$x^{-2}$

x^2

$x^{-3}$

It's impossible because the value is negative.

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 #21 : How To Simplify Square Roots

Simplify: $\sqrt{8^7}$

Possible Answers:

$8^6\sqrt{2}$

$128\sqrt{2}$

$8^3\sqrt{8}$

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

$64\sqrt{2}$

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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Example Question #31 : How To Simplify Square Roots

Simplify: $\sqrt{(-9)^6}$

Possible Answers:

$\pm729$

$\frac{1}{729}$

729

$-\frac{1}{729}$

Correct answer:

729

Explanation:

To solve this problem, consider PEMDAS first. Do the math inside the parentheses before performing the square root.

$\sqrt{(-9)^6}=\sqrt{9^6}$ Then we just find perfect squares inside the radical.

$\sqrt{9^6}=\sqrt{9^2}*\sqrt{9^2}*\sqrt{9^2}=9*9*9=729$

Square roots will always generate real values.

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Example Question #62 : Basic Squaring / Square Roots

Simplify: $\sqrt{(-12)^4}$

Possible Answers:

144

$\frac{1}{144}$

$\pm144$

$-\frac{1}{144}$

Correct answer:

144

Explanation:

To solve this problem, consider PEMDAS first. Do the math inside the parentheses before performing the square root.

$\sqrt{(-12)^4}=\sqrt{12^4}$ Then we just find perfect squares inside the radical.

$\sqrt{12^4}=\sqrt{12^2}*\sqrt{12^2}=12*12=144$

Square roots will always generate real values.

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Example Question #31 : How To Simplify Square Roots

Simplify: $\sqrt{8x^5}$

Possible Answers:

$4x^4\sqrt{2x}$

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

$2\sqrt{x^3}$

$8x^2\sqrt{x}$

$2x\sqrt{2x}$

Correct answer:

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

Explanation:

To simplify radicals, we need to find a perfect square to factor out. In this case, its 4x^4 .

$\sqrt{8x^5}=\sqrt{4x^4}*\sqrt{2x}=2x^2\sqrt{2x}$

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Example Question #32 : How To Simplify Square Roots

Simplify: $\sqrt{x^{4n+1}}$

Possible Answers:

$x^{2n+1}\sqrt{x}$

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

$x^n\sqrt{x}$

$x^n\sqrt{x^{n+1}}$

$x^n\sqrt{x^n}$

Correct answer:

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

Explanation:

We need to apply the power rule of multiplying exponents and break it down.

$\sqrt{x^{4n+1}}=\sqrt{(x^{2n})^2*x^1}=x^{2n}\sqrt{x}$

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Example Question #33 : How To Simplify Square Roots

Simplify: $\frac{\sqrt{88}}{8}$

Possible Answers:

$\frac{\sqrt{22}}{6}$

$\frac{\sqrt{22}}{2}$

$\frac{\sqrt{22}}{4}$

$\frac{\sqrt{11}}{2}$

$\frac{\sqrt{11}}{4}$

Correct answer:

$\frac{\sqrt{22}}{4}$

Explanation:

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

$\frac{\sqrt{88}}{8}=\frac{\sqrt{4}*\sqrt{22}}{8}=\frac{2\sqrt{22}}{8}=\frac{\sqrt{22}}{4}$

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Example Question #34 : How To Simplify Square Roots

Simplify: $\frac{\sqrt{x^5y^4}}{xy^2}$

Possible Answers:

$\frac{x^2\sqrt{x}}{y}$

$x\sqrt{x}$

$\frac{x\sqrt{x}}{y}$

$\frac{x}{y}$

$\frac{\sqrt{xy}}{xy}$

Correct answer:

$x\sqrt{x}$

Explanation:

To simplify radicals, we need to find a perfect square to factor out. In this case, its x^4y^4 .

$\frac{\sqrt{x^5y^4}}{xy^2}=\frac{\sqrt{x^4y^4}*\sqrt{x}}{xy^2}=\frac{x^2y^2\sqrt{x}}{xy^2}={x\sqrt{x}}$

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Example Question #35 : How To Simplify Square Roots

Simplify: $\sqrt{x^3y^2z}$

Possible Answers:

$x^2y^2\sqrt{z}$

$x^2y\sqrt{xz}$

$xyz\sqrt{x}$

$xy\sqrt{xz}$

$xz\sqrt{xy}$

Correct answer:

$xy\sqrt{xz}$

Explanation:

To simplify radicals, we need to find a perfect square to factor out. In this case, its x^2y^2 .

$\sqrt{x^3y^2z}=\sqrt{x^2y^2}*\sqrt{xz}=xy\sqrt{xz}$

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Example Question #32 : How To Simplify Square Roots

Simplify: $\frac{\sqrt{75}}{10}$

Possible Answers:

$\frac{\sqrt{5}}{2}$

$\frac{\sqrt{3}}{5}$

$\frac{\sqrt{3}}{3}$

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

$\frac{\sqrt{5}}{5}$

Correct answer:

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

Explanation:

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

$\frac{\sqrt{75}}{10}=\frac{\sqrt{25}*\sqrt{3}}{10}=\frac{5\sqrt{3}}{10}=\frac{\sqrt{3}}{2}$

Remember you can reduce the fraction even further.

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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #61 : Basic Squaring / Square Roots

Example Question #21 : How To Simplify Square Roots

Example Question #31 : How To Simplify Square Roots

Example Question #62 : Basic Squaring / Square Roots

Example Question #31 : How To Simplify Square Roots

Example Question #32 : How To Simplify Square Roots

Example Question #33 : How To Simplify Square Roots

Example Question #34 : How To Simplify Square Roots

Example Question #35 : How To Simplify Square Roots

Example Question #32 : How To Simplify Square Roots

All SAT Math Resources

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