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SAT Math : Simplifying Square Roots

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Example Questions

Example Question #41 : How To Simplify Square Roots

Simplify: $\sqrt{448}$ $\displaystyle \sqrt{448}$

Possible Answers:

$8\sqrt{7}$ $\displaystyle 8\sqrt{7}$

$7\sqrt{8}$ $\displaystyle 7\sqrt{8}$

$12\sqrt{7}$ $\displaystyle 12\sqrt{7}$

$64\sqrt{7}$ $\displaystyle 64\sqrt{7}$

$16\sqrt{7}$ $\displaystyle 16\sqrt{7}$

Correct answer:

$8\sqrt{7}$ $\displaystyle 8\sqrt{7}$

Explanation:

To simplify square roots, we need to factor out perfect squares. In this case, it's $\displaystyle 64$.

$\sqrt{448}=\sqrt{64}*\sqrt{7}=8\sqrt{7}$ $\displaystyle \sqrt{448}=\sqrt{64}*\sqrt{7}=8\sqrt{7}$

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Example Question #42 : How To Simplify Square Roots

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

Possible Answers:

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

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

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

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

$x\sqrt{x}$ $\displaystyle x\sqrt{x}$

Correct answer:

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

Explanation:

To simplify the radical, we should factor out perfect squares.

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

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Example Question #43 : How To Simplify Square Roots

Simplify: $\sqrt{x^7y^3}$ $\displaystyle \sqrt{x^7y^3}$

Possible Answers:

$x^6y^2\sqrt{xy}$ $\displaystyle x^6y^2\sqrt{xy}$

$x^3y\sqrt{xy}$ $\displaystyle x^3y\sqrt{xy}$

$xy\sqrt{x^3y^2}$ $\displaystyle xy\sqrt{x^3y^2}$

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

$x^3\sqrt{xy^3}$ $\displaystyle x^3\sqrt{xy^3}$

Correct answer:

$x^3y\sqrt{xy}$ $\displaystyle x^3y\sqrt{xy}$

Explanation:

To simplify the square roots, we need to factor out the perfect squares.

$\sqrt{x^7y^3}=\sqrt{x^2}*\sqrt{x^2}*\sqrt{x^2}*\sqrt{y^2}*\sqrt{xy}=x^3y\sqrt{xy}$ $\displaystyle \sqrt{x^7y^3}=\sqrt{x^2}*\sqrt{x^2}*\sqrt{x^2}*\sqrt{y^2}*\sqrt{xy}=x^3y\sqrt{xy}$

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Example Question #44 : How To Simplify Square Roots

Simplify: $\sqrt{4^5*5^4}$ $\displaystyle \sqrt{4^5*5^4}$

Possible Answers:

$2^4*5^4\sqrt{2}$ $\displaystyle 2^4*5^4\sqrt{2}$

$2^4*5^2\sqrt{10}$ $\displaystyle 2^4*5^2\sqrt{10}$

$2^3*5^2\sqrt{32}$ $\displaystyle 2^3*5^2\sqrt{32}$

2^5*5^2 $\displaystyle 2^5*5^2$

$2^5*5^2\sqrt{5}$ $\displaystyle 2^5*5^2\sqrt{5}$

Correct answer:

2^5*5^2 $\displaystyle 2^5*5^2$

Explanation:

To simplify the square roots, we need to factor out perfect squares.

$\sqrt{4^5*5^4}=\sqrt{4^2}*\sqrt{4^2}*\sqrt{4}*\sqrt{5^2}*\sqrt{5^2}=4*4*2*5*5$ $\displaystyle \sqrt{4^5*5^4}=\sqrt{4^2}*\sqrt{4^2}*\sqrt{4}*\sqrt{5^2}*\sqrt{5^2}=4*4*2*5*5$

We can combine to have two different bases. Remember 4=2^2 $\displaystyle 4=2^2$.

$\displaystyle 4*4*2*5*5=2^2*2^2*2*5^2=2^5*5^2$

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Example Question #44 : How To Simplify Square Roots

Simplify: $\sqrt{(-8)^8}$ $\displaystyle \sqrt{(-8)^8}$

Possible Answers:

-8^4 $\displaystyle -8^4$

$8^{16}$ $\displaystyle 8^{16}$

8^4 $\displaystyle 8^4$

-8^6 $\displaystyle -8^6$

8^8 $\displaystyle 8^8$

Correct answer:

8^4 $\displaystyle 8^4$

Explanation:

To simplify the square root, we need to determine the value of the exponent and then simplify the radical.

$\sqrt{(-8)^8}=\sqrt{8^8}$ $\displaystyle \sqrt{(-8)^8}=\sqrt{8^8}$ Now let's find perfect squares.

$\sqrt{8^8}=\sqrt{8^4}*\sqrt{8^4}=8^2*8^2=8^4$ $\displaystyle \sqrt{8^8}=\sqrt{8^4}*\sqrt{8^4}=8^2*8^2=8^4$

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Example Question #45 : How To Simplify Square Roots

Simplify: $\sqrt{(-9)^{10}}$ $\displaystyle \sqrt{(-9)^{10}}$

Possible Answers:

$9^{10}$ $\displaystyle 9^{10}$

$-9^{-5}$ $\displaystyle -9^{-5}$

$9^{-5}$ $\displaystyle 9^{-5}$

-9^5 $\displaystyle -9^5$

9^5 $\displaystyle 9^5$

Correct answer:

9^5 $\displaystyle 9^5$

Explanation:

To simplify the square root, we need to determine the value of the exponent and then simplify the radical.

$\sqrt{(-9)^{10}}=\sqrt{9^{10}}$ $\displaystyle \sqrt{(-9)^{10}}=\sqrt{9^{10}}$ Now let's find perfect squares.

$\sqrt{9^{10}}=\sqrt{9^5}*\sqrt{9^5}=9^{2}*9^2*\sqrt{9}*\sqrt{9}=9^4*9=9^5$ $\displaystyle \sqrt{9^{10}}=\sqrt{9^5}*\sqrt{9^5}=9^{2}*9^2*\sqrt{9}*\sqrt{9}=9^4*9=9^5$

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Example Question #46 : How To Simplify Square Roots

Simplify: $\frac{\sqrt{48}}{2}$ $\displaystyle \frac{\sqrt{48}}{2}$

Possible Answers:

$\frac{4\sqrt{3}}{3}$ $\displaystyle \frac{4\sqrt{3}}{3}$

$2\sqrt{3}$ $\displaystyle 2\sqrt{3}$

$\sqrt{24}$ $\displaystyle \sqrt{24}$

$4\sqrt{3}$ $\displaystyle 4\sqrt{3}$

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

Correct answer:

$2\sqrt{3}$ $\displaystyle 2\sqrt{3}$

Explanation:

To simplify the radical, we need to find perfect squares. Then if possible, we can reduc the fraction.

$\sqrt{48}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$ $\displaystyle \sqrt{48}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$

$\frac{4\sqrt{3}}{2}=2\sqrt{3}$ $\displaystyle \frac{4\sqrt{3}}{2}=2\sqrt{3}$

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Example Question #141 : Arithmetic

Simplify: $\frac{\sqrt{(-8)^4}}{12}$ $\displaystyle \frac{\sqrt{(-8)^4}}{12}$

Possible Answers:

$-\frac{16}{3}$ $\displaystyle -\frac{16}{3}$

$\frac{8}{3}$ $\displaystyle \frac{8}{3}$

$\frac{64}{3}$ $\displaystyle \frac{64}{3}$

$\frac{16}{3}$ $\displaystyle \frac{16}{3}$

$-\frac{64}{3}$ $\displaystyle -\frac{64}{3}$

Correct answer:

$\frac{16}{3}$ $\displaystyle \frac{16}{3}$

Explanation:

To simplify the radical, let's deal with the parentheses first and apply the exponents. Reduce if necessary.

$\sqrt{(-8)^4}=\sqrt{8^4}=8^2=64$ $\displaystyle \sqrt{(-8)^4}=\sqrt{8^4}=8^2=64$

$\frac{64}{12}=\frac{64\div4}{12\div4}=\frac{16}{3}$ $\displaystyle \frac{64}{12}=\frac{64\div4}{12\div4}=\frac{16}{3}$

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Example Question #41 : How To Simplify Square Roots

Simplify: $\frac{5}{\sqrt{3}+2}$ $\displaystyle \frac{5}{\sqrt{3}+2}$

Possible Answers:

$10-5\sqrt{3}$ $\displaystyle 10-5\sqrt{3}$

$5\sqrt{3}-10$ $\displaystyle 5\sqrt{3}-10$

$\frac{5\sqrt{3}-10}{7}$ $\displaystyle \frac{5\sqrt{3}-10}{7}$

${5\sqrt{3}+10}$ $\displaystyle {5\sqrt{3}+10}$

$\frac{5\sqrt{3}-2}{5}$ $\displaystyle \frac{5\sqrt{3}-2}{5}$

Correct answer:

$10-5\sqrt{3}$ $\displaystyle 10-5\sqrt{3}$

Explanation:

To eliminate a radical expression, we need to multiply top and bottom by the conjugate which is opposite the sign in the expression. Then simplify if necessary.

$\frac{5}{\sqrt{3}+2}=\frac{5}{\sqrt{3}+2}*\frac{\sqrt{3}-2}{\sqrt{3}-2}=\frac{5\sqrt{3}-10}{3-4}=-(5\sqrt{3}-10)=10-5\sqrt{3}$ $\displaystyle \frac{5}{\sqrt{3}+2}=\frac{5}{\sqrt{3}+2}*\frac{\sqrt{3}-2}{\sqrt{3}-2}=\frac{5\sqrt{3}-10}{3-4}=-(5\sqrt{3}-10)=10-5\sqrt{3}$

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Example Question #50 : How To Simplify Square Roots

Simplify:

$\sqrt{6^5*8^3}$ $\displaystyle \sqrt{6^5*8^3}$

Possible Answers:

$576\sqrt{3}$ $\displaystyle 576\sqrt{3}$

$24\sqrt{3}$ $\displaystyle 24\sqrt{3}$

$6^4*2^6\sqrt{48}$ $\displaystyle 6^4*2^6\sqrt{48}$

$576\sqrt{6}$ $\displaystyle 576\sqrt{6}$

$1152\sqrt{3}$ $\displaystyle 1152\sqrt{3}$

Correct answer:

$1152\sqrt{3}$ $\displaystyle 1152\sqrt{3}$

Explanation:

To simplify square roots, we need to find perfect squares to factor out.

$\sqrt{6^5*8^3}=\sqrt{6^2}*\sqrt{6^2}*\sqrt{8^2}*\sqrt{48}$ $\displaystyle \sqrt{6^5*8^3}=\sqrt{6^2}*\sqrt{6^2}*\sqrt{8^2}*\sqrt{48}$

We can also simplify $\sqrt{48}=\sqrt{16}\times \sqrt{3}=4\sqrt{3}$ $\displaystyle \sqrt{48}=\sqrt{16}\times \sqrt{3}=4\sqrt{3}$.

Thus,

$6^2\cdot 8\cdot4\sqrt{3}$ $\displaystyle 6^2\cdot 8\cdot4\sqrt{3}$

We can compute the numbers outside to get a final answer of $1152\sqrt{3}$ $\displaystyle 1152\sqrt{3}$.

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SAT Math : Simplifying Square Roots

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #41 : How To Simplify Square Roots

Example Question #42 : How To Simplify Square Roots

Example Question #43 : How To Simplify Square Roots

Example Question #44 : How To Simplify Square Roots

Example Question #44 : How To Simplify Square Roots

Example Question #45 : How To Simplify Square Roots

Example Question #46 : How To Simplify Square Roots

Example Question #141 : Arithmetic

Example Question #41 : How To Simplify Square Roots

Example Question #50 : How To Simplify Square Roots

All SAT Math Resources

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