Basic Squaring / Square Roots - GRE Quantitative Reasoning

Card 0 of 528

Question

Solve for :

$\sqrt{\frac{x}{5}}=\sqrt{30}$

Answer

You can "break apart" the fractional square root:

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

into:

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

Therefore, you can rewrite your equation as:

$\frac{\sqrt{x}}{\sqrt{5}} =\sqrt{30}$

Now, multiply both sides by $\sqrt{5}$ :

$\sqrt{x}=\sqrt{5}\sqrt{30}=\sqrt{150}$

Now, you can square both sides of your equation and get:

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Question

$\sqrt{45}\cdot \sqrt{3}=?$

Answer

Begin by multiplying what is under the square roots together. Remember:

$\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$

Therefore:

$\sqrt{45}\cdot \sqrt{3}=\sqrt{45\cdot 3}=\sqrt{135}$

We must now try to reduce the square root. Look for factors that can be taken out that can be easily square-rooted. It's a good idea to start with small square roots $\sqrt{4},\sqrt{9}, etc.$ . In this case, we can factor out a 9 from 135:

$\sqrt{135}=\sqrt{9\cdot 15}=\sqrt{9}\cdot \sqrt{15}$

Now take the square root of 9, and we are left with our answer:

$3\sqrt{15}$

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Question

The length of a square courtyard is $\sqrt{34}$ feet.

What is the area of the courtyard?

Answer

The area of a square is length squared. In this case, that would be $(\sqrt{34})^2$ which is equivalent to $\sqrt{34}\cdot\sqrt{34}$ . At this point, you multiply the values underneath the square root, then simplify:

$\sqrt{34}\cdot\sqrt{34}=\sqrt{34^2}$ .

The square root of a base squared is then just the base:

$\sqrt{34^2}=34.$

Lastly, we apply the appropriate units, therefore the area of the courtyard is .

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Question

Solve: $\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}$

Answer

To solve this problem, we must know how to multiply/divide with square roots. The rules involved are quite simple. When multiplying square roots together, multiply the coeffcients of the two numbers together and then multiply the two radicands together. The product of the radicand becomes the new radicand and the product of the coeffcients become the new coeffcient. The radicand is simply the number within the square root.

$\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}=\frac{40\sqrt{12}}{3\sqrt{3}}$

Dividing square roots follows the exact same rules as multiplying square roots, divide the coeffcients together, the result is the coeffcient for final answer. Do the same for the radicands.

$\frac{40\sqrt{12}}{3\sqrt{3}}=\frac{40\sqrt{4}}{3}=\frac{80}{3}$

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Question

Solve for :

$\sqrt{\frac{x}{5}}=\sqrt{30}$

Answer

You can "break apart" the fractional square root:

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

into:

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

Therefore, you can rewrite your equation as:

$\frac{\sqrt{x}}{\sqrt{5}} =\sqrt{30}$

Now, multiply both sides by $\sqrt{5}$ :

$\sqrt{x}=\sqrt{5}\sqrt{30}=\sqrt{150}$

Now, you can square both sides of your equation and get:

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Question

$\sqrt{45}\cdot \sqrt{3}=?$

Answer

Begin by multiplying what is under the square roots together. Remember:

$\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$

Therefore:

$\sqrt{45}\cdot \sqrt{3}=\sqrt{45\cdot 3}=\sqrt{135}$

We must now try to reduce the square root. Look for factors that can be taken out that can be easily square-rooted. It's a good idea to start with small square roots $\sqrt{4},\sqrt{9}, etc.$ . In this case, we can factor out a 9 from 135:

$\sqrt{135}=\sqrt{9\cdot 15}=\sqrt{9}\cdot \sqrt{15}$

Now take the square root of 9, and we are left with our answer:

$3\sqrt{15}$

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Question

The length of a square courtyard is $\sqrt{34}$ feet.

What is the area of the courtyard?

Answer

The area of a square is length squared. In this case, that would be $(\sqrt{34})^2$ which is equivalent to $\sqrt{34}\cdot\sqrt{34}$ . At this point, you multiply the values underneath the square root, then simplify:

$\sqrt{34}\cdot\sqrt{34}=\sqrt{34^2}$ .

The square root of a base squared is then just the base:

$\sqrt{34^2}=34.$

Lastly, we apply the appropriate units, therefore the area of the courtyard is .

Compare your answer with the correct one above

Question

Solve: $\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}$

Answer

To solve this problem, we must know how to multiply/divide with square roots. The rules involved are quite simple. When multiplying square roots together, multiply the coeffcients of the two numbers together and then multiply the two radicands together. The product of the radicand becomes the new radicand and the product of the coeffcients become the new coeffcient. The radicand is simply the number within the square root.

$\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}=\frac{40\sqrt{12}}{3\sqrt{3}}$

Dividing square roots follows the exact same rules as multiplying square roots, divide the coeffcients together, the result is the coeffcient for final answer. Do the same for the radicands.

$\frac{40\sqrt{12}}{3\sqrt{3}}=\frac{40\sqrt{4}}{3}=\frac{80}{3}$

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Question

Solve for :

$\sqrt{\frac{x}{5}}=\sqrt{30}$

Answer

You can "break apart" the fractional square root:

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

into:

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

Therefore, you can rewrite your equation as:

$\frac{\sqrt{x}}{\sqrt{5}} =\sqrt{30}$

Now, multiply both sides by $\sqrt{5}$ :

$\sqrt{x}=\sqrt{5}\sqrt{30}=\sqrt{150}$

Now, you can square both sides of your equation and get:

Compare your answer with the correct one above

Question

$\sqrt{45}\cdot \sqrt{3}=?$

Answer

Begin by multiplying what is under the square roots together. Remember:

$\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$

Therefore:

$\sqrt{45}\cdot \sqrt{3}=\sqrt{45\cdot 3}=\sqrt{135}$

We must now try to reduce the square root. Look for factors that can be taken out that can be easily square-rooted. It's a good idea to start with small square roots $\sqrt{4},\sqrt{9}, etc.$ . In this case, we can factor out a 9 from 135:

$\sqrt{135}=\sqrt{9\cdot 15}=\sqrt{9}\cdot \sqrt{15}$

Now take the square root of 9, and we are left with our answer:

$3\sqrt{15}$

Compare your answer with the correct one above

Question

The length of a square courtyard is $\sqrt{34}$ feet.

What is the area of the courtyard?

Answer

The area of a square is length squared. In this case, that would be $(\sqrt{34})^2$ which is equivalent to $\sqrt{34}\cdot\sqrt{34}$ . At this point, you multiply the values underneath the square root, then simplify:

$\sqrt{34}\cdot\sqrt{34}=\sqrt{34^2}$ .

The square root of a base squared is then just the base:

$\sqrt{34^2}=34.$

Lastly, we apply the appropriate units, therefore the area of the courtyard is .

Compare your answer with the correct one above

Question

Solve: $\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}$

Answer

To solve this problem, we must know how to multiply/divide with square roots. The rules involved are quite simple. When multiplying square roots together, multiply the coeffcients of the two numbers together and then multiply the two radicands together. The product of the radicand becomes the new radicand and the product of the coeffcients become the new coeffcient. The radicand is simply the number within the square root.

$\frac{5\sqrt{2}*8\sqrt{6}}{3\sqrt{3}}=\frac{40\sqrt{12}}{3\sqrt{3}}$

Dividing square roots follows the exact same rules as multiplying square roots, divide the coeffcients together, the result is the coeffcient for final answer. Do the same for the radicands.

$\frac{40\sqrt{12}}{3\sqrt{3}}=\frac{40\sqrt{4}}{3}=\frac{80}{3}$

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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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Question

Simplify $\sqrt{a^{3}b^{4}c^{5}}$ .

Answer

Rewrite what is under the radical in terms of perfect squares:

$x^{2}=x\cdot x$

$x^{4}=x^{2}\cdot x^{2}$

$x^{6}=x^{3}\cdot x^{3}$

Therefore, $\sqrt{a^{3}b^{4}c^{5}}= \sqrt{a^{2}a^{1}b^{4}c^{4}c^{1}}=ab^{2}c^{2}\sqrt{ac}$ .

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Question

Which of the following is equivalent to $\frac{x + \sqrt{3}}{3x + \sqrt{2}}$ ?

Answer

Multiply by the conjugate and the use the formula for the difference of two squares:

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

$\frac{x + \sqrt{3}}{3x + \sqrt{2}}\cdot \frac{3x - \sqrt{2}}{3x - \sqrt{2}}$

$\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{(3x)^{2} - (\sqrt{2})^{2}}$

$\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}$

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