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ACT Math : Basic Squaring / Square Roots

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Example Questions

Example Question #31 : Arithmetic

Solve for :

$\frac{1}{2}x\sqrt[4]{9072} + x\sqrt[3]{54} = 18$

Possible Answers:

$x = \frac{6}{(\sqrt[3]{7} + \sqrt[4]{2})}$

$x = -\frac{6}{(\sqrt[4]{2} - \sqrt[3]{7})}$

$x = \frac{3}{(\sqrt[8]{7} + \sqrt[6]{2})}$

$x = \frac{6}{(\sqrt[4]{7} + \sqrt[3]{2})}$

$x = -\frac{6}{(\sqrt[4]{7} + \sqrt[3]{2})}$

Correct answer:

$x = \frac{6}{(\sqrt[4]{7} + \sqrt[3]{2})}$

Explanation:

To begin solving this problem, find the greatest perfect square for all quantities under a radical. $3x\sqrt[4]{9072}$ might seem intimidating, but remember that raising even single-digit numbers to the fourth power creates huge numbers. In this case, 9072 is divisible by 1296 , a perfect fourth power.

$\frac{1}{2}x\sqrt[4]{9072} + x\sqrt[3]{54} = 18$ ---> $\frac{1}{2}x\sqrt[4]{7\cdot 1296} + x\sqrt[3]{2\cdot 27} = 18$

Pull the perfect terms out of each term on the left:

$\frac{1}{2}x\sqrt[4]{7\cdot 1296} + x\sqrt[3]{2\cdot 27} = 18$ ---> $3x\sqrt[4]{7} + 3x\sqrt[3]{2} = 18$

Next, factor out from the left-hand side:

$3x\sqrt[4]{7} + 3x\sqrt[3]{2} = 18$ ---> $3x(\sqrt[4]{7} + \sqrt[3]{2}) = 18$

Lastly, isolate , remembering to simplify the fraction where possible:

$3x(\sqrt[4]{7} + \sqrt[3]{2}) = 18$ ---> $x = \frac{6}{(\sqrt[4]{7} + \sqrt[3]{2})}$

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Example Question #31 : Arithmetic

Simplify: $\sqrt{384} + \sqrt{216}$

Possible Answers:

$16\sqrt{7}$

$14\sqrt{6}$

$8\sqrt{8}$

$12\sqrt{7}$

$8\sqrt{14}$

Correct answer:

$14\sqrt{6}$

Explanation:

To start, begin pulling the largest perfect square you can out of each number:

$\sqrt{384} = \sqrt {4 \cdot 4 \cdot 4 \cdot 6 } = \sqrt {64 \cdot 6} = 8\sqrt 6$

$\sqrt {216} = \sqrt {6 \cdot 6 \cdot 6} = \sqrt {36 \cdot 6} = 6\sqrt 6$

So, $\sqrt{384} + \sqrt{216} = 8\sqrt 6 + 6\sqrt 6 = 14\sqrt{6}$ . You can just add the two terms together once they have a common radical.

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Example Question #31 : Arithmetic

Simplify: $\sqrt{6300} - \sqrt{700}$

Possible Answers:

$20 \sqrt{7}$

$7\sqrt{13}$

$15\sqrt{21}$

$8\sqrt{15}$

$13\sqrt{13}$

Correct answer:

$20 \sqrt{7}$

Explanation:

Again here, it is easiest to recognize that both of our terms are divisible by , a prime number likely to appear in our final answer:

$\sqrt{6300} - \sqrt{700} = \sqrt{7\cdot 900} - \sqrt{7 \cdot 100}$

Now, simplify our perfect squares:

$\sqrt{7\cdot 900} - \sqrt{7 \cdot 100} = 30\sqrt 7 - 10 \sqrt 7$

Lastly, subtract our terms with a common radical:

$30\sqrt{7} - 10\sqrt {7} = 20\sqrt{7}$

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Example Question #31 : Arithmetic

Solve for : $x\sqrt{45} + x\sqrt{108} = 12$

Possible Answers:

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

$x = \frac{4}{(\sqrt{5} +\sqrt{12})}$

$x = (\sqrt{5} +2\sqrt{3})$

$x = \frac{12}{(\sqrt{5} +2\sqrt{3})}$

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

Correct answer:

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

Explanation:

To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

$x\sqrt{45} + x\sqrt{108} = 12$ ---> $x\sqrt{5 \cdot 9} + x\sqrt{3\cdot 36 } = 12$

Factor out the square root of each perfect square:

$x\sqrt{5 \cdot 9} + x\sqrt{3 \cdot 36} = 12$ ---> $3x\sqrt{5} + 6x\sqrt{3} = 12$

Next, factor out from each term on the left-hand side of the equation:

$3x\sqrt{5} + 3x\sqrt{12} = 12$ ---> $3x(\sqrt{5} +2\sqrt{3}) = 12$

Lastly, isolate :

$3x(\sqrt{5} +2\sqrt{3}) = 12$ ---> $x = \frac{4}{(\sqrt{5} +2\sqrt{3})}$

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Example Question #1 : Square Roots And Operations

Find the product:

$\sqrt{50}\cdot \sqrt{162}$

Possible Answers:

$8\sqrt{7}$

$\sqrt{71}$

$\sqrt{63}$

Correct answer:

Explanation:

Simplify the radicals, then multiply:

$\sqrt{25\cdot 2}\cdot \sqrt{81\cdot 2}=5\sqrt{2}\cdot9\sqrt{2}=45\cdot2=90$

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Example Question #2 : Square Roots And Operations

Simplify the following completely:

$\sqrt{3}\cdot\sqrt{2}\cdot\sqrt{10}$

Possible Answers:

$2\sqrt{15}$

$\sqrt{15}$

$\sqrt{60}$

Correct answer:

$2\sqrt{15}$

Explanation:

To simplify this expression, simply multiply the radicands and reduce to simplest form.

$\sqrt{3}\cdot\sqrt{2}\cdot\sqrt{10}\rightarrow \sqrt{3\cdot2\cdot10}\rightarrow \sqrt{60}$

$\sqrt{60}\rightarrow \sqrt{4\cdot15}\rightarrow \sqrt{2^2\cdot15}\rightarrow 2\sqrt{15}$

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Example Question #1 : How To Multiply Square Roots

Simplify:

$\small \sqrt{30} * \sqrt{462} * \sqrt{22}$

Possible Answers:

$\small 24\sqrt{21}$

$\small 66\sqrt{70}$

$\small 14\sqrt{330}$

$\small 80\sqrt{14}$

$\small 105\sqrt{7}$

Correct answer:

$\small 66\sqrt{70}$

Explanation:

When multiplying square roots, the easiest thing to do is first to factor each root. Thus:

$\small \small \sqrt{30} * \sqrt{462} * \sqrt{22} = \sqrt{2*3*5} * \sqrt{2*3*7*11} * \sqrt{2*11}$

Now, when you combine the multiplied roots, it will be easier to come to your final solution. Just multiply together everything "under" the roots:

$\small \small \small \sqrt{2*3*5} * \sqrt{2*3*7*11} * \sqrt{2*11} = \sqrt{2*2*2*3*3*5*7*11*11}$

Finally this can be simplified as:

$\small \small \small \sqrt{2*2*2*3*3*5*7*11*11} = 2*3*11\sqrt{2*5*7} = 66\sqrt{70}$

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Example Question #4 : Square Roots And Operations

Simplify the following:

$\small \sqrt{140} * \sqrt{21} * \sqrt{10}$

Possible Answers:

$\small 42\sqrt{5}$

$\small 21\sqrt{10}$

$\small 35\sqrt{14}$

$\small 10\sqrt{21}$

$70\sqrt{6}$

Correct answer:

$70\sqrt{6}$

Explanation:

When multiplying square roots, the easiest thing to do is first to factor each root. Thus:

$\small \sqrt{140} * \sqrt{21} * \sqrt{10}=\sqrt{2*2*5*7} * \sqrt{3*7} * \sqrt{2*5}$

Now, when you combine the multiplied roots, it will be easier to come to your final solution. Remember that multiplying roots is very easy! Just multiply together everything "under" the roots:

$\small \sqrt{2*2*5*7} * \sqrt{3*7} * \sqrt{2*5} = \sqrt{2*2*2*3*5*5*7*7}$

Finally this can be simplified as:

$\small \sqrt{2*2*3*5*5*7*7}=2*5*7\sqrt{2*3}=70\sqrt{2*3}$

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Example Question #5 : Square Roots And Operations

State the product: $5\sqrt{15} \cdot 8\sqrt{6}$

Possible Answers:

$13\sqrt{40}$

$40\sqrt{90}$

$40\sqrt{21}$

$120\sqrt{10}$

$35\sqrt{18}$

Correct answer:

$120\sqrt{10}$

Explanation:

Don't try to do too much at first for this problem. Multiply your radicals and your coefficients, then worry about any additional simplification.

$5\sqrt{15} \cdot 8\sqrt{6} = 40\sqrt{90}$

Now simplify the radical.

$40\sqrt{90} = 40\sqrt {9 \cdot 10} = 120 \sqrt{10}$

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Example Question #6 : Square Roots And Operations

Find the product: $6\sqrt{14} \cdot 3\sqrt{7}$

Possible Answers:

$9\sqrt{21}$

$126\sqrt{2}$

$113\sqrt{31}$

$20\sqrt{97}$

$18\sqrt{98}$

Correct answer:

$126\sqrt{2}$

Explanation:

Don't try to do too much at first for this problem. Multiply your radicals and your coefficients, then worry about any additional simplification.

$6\sqrt{14} \cdot 3\sqrt{7} = 18\sqrt{98}$

Now, simplify your radical.

$18\sqrt{98} = 18\sqrt{49 \cdot 2} = 126\sqrt{2}$

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ACT Math : Basic Squaring / Square Roots

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #31 : Arithmetic

Example Question #31 : Arithmetic

Example Question #31 : Arithmetic

Example Question #31 : Arithmetic

Example Question #1 : Square Roots And Operations

Example Question #2 : Square Roots And Operations

Example Question #1 : How To Multiply Square Roots

Example Question #4 : Square Roots And Operations

Example Question #5 : Square Roots And Operations

Example Question #6 : Square Roots And Operations

All ACT Math Resources

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