Basic Squaring / Square Roots - GRE Quantitative Reasoning

In order to determine the relationship between Quantity A and Quantity B, let's convert both to square roots. In order to do this, we must square Quantity A so it becomes √81 which is equivalent to 9. Now to Quantity B, we must simplify by adding the two values together (25 + 55) to get √80.

√81 is greater than the √80 because 81 is greater than 80. Thus Quantity A is greater.

Since is between 10 and 20, it can be any real number between 100 and 400. Therefore, the relationship cannot be determined since could fall anywhere between these two limits, including between 399 and 400.

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:

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 (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:

But 243 can also be factored:

Following the same principle as for the 100, the problem would become
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 9*10*3.
9 and 10 may be multiplied together, yielding the final simplified answer of

To solve the equation , we can first factor the numbers under the square roots.

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

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

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:

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:

Note:

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.

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.

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:

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:

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:

However,

Thus,

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:

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:

Use the following arithmetic steps to reduce .

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 can be reduced.

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:

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:

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

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

First find all of the prime factors of

So