Simplifying Square Roots - GRE Quantitative Reasoning

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Question

Simplify:

$\sqrt[2]{24,300}$

Answer

$\sqrt[2]{24,300}$

When simplifying the square root of a number that may not have a whole number root, it's helpful to approach the problem by finding common factors of the number inside the radicand. In this case, the number is 24,300.

What are the factors of 24,300?

24,300 can be factored into:

$24,300: 243\cdot 100$

When there are factors that appear twice, they may be pulled out of the radicand. For instance, 100 is a multiple of 24,300. When 100 is further factored, it is $10^{2}$ (or 10x10). However, 100 wouldn't be pulled out of the radicand, but the square root of 100 because the square root of 24,300 is being taken. The 100 is part of the24,300. This means that the problem would be rewritten as: $10\sqrt[2]{243}$

But 243 can also be factored: $24,300: (3\cdot 81)\cdot100$
$24,300: {\color{Orange} 9}\cdot{\color{Orange} 9}\cdot {\color{Cyan} 10}\cdot {\color{cyan} 10}\cdot 3$
Following the same principle as for the 100, the problem would become
$9\cdot 10\sqrt[2]{3}$ because there is only one factor of 3 left in the radicand. If there were another, the radicand would be lost and it would be 9103.
9 and 10 may be multiplied together, yielding the final simplified answer of
$90\sqrt[2]{3}$

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Question

$\sqrt{180} + \sqrt{125} = ?$

Answer

To solve the equation $\sqrt{180} + \sqrt{125} = ?$ , we can first factor the numbers under the square roots.

$\sqrt{2\cdot 2\cdot 3\cdot 3\cdot 5} + \sqrt{5\cdot 5\cdot 5} = ?$

When a factor appears twice, we can take it out of the square root.

$2\cdot 3\sqrt{5} + 5\sqrt{5} = ?$

$6\sqrt{5} + 5\sqrt{5} = ?$

Now the numbers can be added directly because the expressions under the square roots match.

$6\sqrt{5} + 5\sqrt{5} = 11\sqrt{5}$

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Question

Simplify.

$\sqrt{624}$

Answer

To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 624 and is also a perfect square.

Therefore we can rewrite the square root of 624 as:

$\sqrt{624}=\sqrt{16\times 39}=\sqrt{16}\sqrt{39}=4\sqrt{39}$

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Question

Reduce $\sqrt{400}$ to its simplest form.

Answer

To simplify, we must try to find factors which are perfect squares. In this case 20 is a factor of 400 and is also a perfect square.

Thus we can rewrite the problem as:

$\sqrt{400}=\sqrt{20\times 20}=20$

Note: $20^{2}=20\times20=400$

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Question

Simplify.

$\sqrt{720}$

Answer

Use the following steps to reduce this square root.

To simplify, we must try to find factors which are perfect squares. In this case 144 is a factor of 720 and is also a perfect square.

Thus we can rewrite the problem as follows.

$\sqrt{720}=\sqrt{144\times 5}=\sqrt{144}\sqrt{5}=12\sqrt{5}$

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Question

Find the square root of .

Answer

Use the following steps to find the square root of

To simplify, we must try to find factors which are perfect squares. In this case 900 is a factor of 1800 and is also a perfect square.

Thus we can rewrite the problem as follows.

$\sqrt{1,800}=\sqrt{900\times 2}=\sqrt{900}\sqrt{2}=30\sqrt{2}$

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Question

Simplify.

$\sqrt{54}$

Answer

To simplify, we must try to find factors which are perfect squares. In this case 9 is a factor of 54 and is also a perfect square.

To reduce this expression, use the following steps:

$\sqrt{54}=\sqrt{9\times 6}=\sqrt{9}\sqrt{6}=3\sqrt{6}$

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Question

Reduce.

$\sqrt{72}$

Answer

To simplify, we must try to find factors which are perfect squares. In this case 36 is a factor of 72 and is also a perfect square.

To reduce this expression, use the following arithmetic steps:

$\sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\sqrt{2}=6\sqrt{2}$

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Question

Which quantity is greater: $\sqrt{900}$ or ?

Answer

To simplify, we must try to find factors which are perfect squares. In this case 30 is a factor of 900 and is also a perfect square.

The square root of is equal to:

$\sqrt{900}=\sqrt{30\times 30}=30$

However,

$\sqrt{900}=\sqrt{-30\times -30}=-30$

Thus, $\pm30$ $=\sqrt{900}$

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Question

Reduce.

$\sqrt{32}$

Answer

To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 32 and is also a perfect square.

To reduce this expression, use the following steps:

$\sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\sqrt{2}=4\sqrt{2}$

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Question

Find the square root of .

Answer

To simplify, we must try to find factors which are perfect squares. In this case 4 is a factor of 164 and is also a perfect square.

To find the square root of , use the following steps:

$\sqrt{164}=\sqrt{4\times 41}=\sqrt{41}\sqrt{4}=2\sqrt{41}$

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Question

Reduce.

$\sqrt{192}$

Answer

Use the following arithmetic steps to reduce $\sqrt{192}$ .

To simplify, we must try to find factors which are perfect squares. In this case 64 is a factor of 192 and is also a perfect square.

Note and are both factors of , however only $\sqrt{64}$ can be reduced.

$\sqrt{192}=\sqrt{64\times 3}=\sqrt{64}\sqrt{3}=8\sqrt{3}$

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Question

Reduce.

$\sqrt{368}$

Answer

To reduce this expression, first find factors of , then reduce.

To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 368 and is also a perfect square.

The solution is:

$\sqrt{368}=\sqrt{16\times 23}=\sqrt{16}\sqrt{23}=4\sqrt{23}$

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Question

Find the square root of .

Answer

To reduce this expression, first find factors of , then reduce.

To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 416 and is also a perfect square.

The solution is:

$\sqrt{416}=\sqrt{16\times 26}=\sqrt{16}\sqrt{26}=4\sqrt{26}$

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Question

Simplify the following: (√(6) + √(3)) / √(3)

Answer

Begin by multiplying top and bottom by √(3):

(√(18) + √(9)) / 3

Note the following:

√(9) = 3

√(18) = √(9 * 2) = √(9) * √(2) = 3 * √(2)

Therefore, the numerator is: 3 * √(2) + 3. Factor out the common 3: 3 * (√(2) + 1)

Rewrite the whole fraction:

(3 * (√(2) + 1)) / 3

Simplfy by dividing cancelling the 3 common to numerator and denominator: √(2) + 1

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Question

what is

√0.0000490

Answer

easiest way to simplify: turn into scientific notation

√0.0000490= √4.9 X 10-5

finding the square root of an even exponent is easy, and 49 is a perfect square, so we can write out an improper scientific notation:

√4.9 X 10-5 = √49 X 10-6

√49 = 7; √10-6 = 10-3 this is equivalent to raising 10-6 to the 1/2 power, in which case all that needs to be done is multiply the two exponents: 7 X 10-3= 0.007

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Question

Which of the following is the most simplified form of:

$\sqrt{468}$

Answer

First find all of the prime factors of

$468=6\ast78=6\ast6\ast13=2\ast3\ast2\ast3\ast13$

So $\sqrt{468}=\sqrt{2\ast2\ast3\ast3\ast13}=2\ast3\sqrt{13}=6\sqrt{13}$

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Question

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

Answer

√75 can be broken down to √25 * √3. Which simplifies to 5√3.

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Question

Simplify: $\sqrt{576}$

Answer

In order to take the square root, divide 576 by 2.

$\dpi{100} \sqrt{576}= \sqrt{2}\sqrt{288}=\sqrt{2}\sqrt{2}\sqrt{144}=\sqrt{4}\sqrt{144}=2\cdot 12=24$

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Question

Simplify $(\frac{16}{81})^{1/4}$ .

Answer

$(\frac{16}{81})^{1/4}$

$\frac{16^{1/4}}{81^{1/4}}$

$\frac{(2\cdot 2\cdot 2\cdot 2)^{1/4}}{(3\cdot 3\cdot 3\cdot 3)^{1/4}}$

$\frac{2}{3}$

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