Square Roots and Operations - GRE Quantitative Reasoning

Begin by breaking down each of the roots in question. Often, for the GRE, your answer arises out of doing such basic "simplification moves".

Quantity A

This is the same as:

, which can be reduced to:

Quantity B

This is the same as:

, which can be reduced to:

Thus, at the end of working through the proper math, you realize that the two values are equal!

Begin by factoring out each of the radicals:

For the first two radicals, you can factor out a or :

The other root values cannot be simply broken down. Now, combine the factors with :

This is your simplest form.

Begin by getting your terms onto the left side of the equation and your numeric values onto the right side of the equation:

Next, you can combine your radicals. You do this merely by subtracting their respective coefficients:

Now, square both sides:

Solve by dividing both sides by :

To solve this problem, we must realize that the only way to add or subtract square roots is if the number under to square root is equivalent to each other. Therefore we must find a way to simplify each square root.

First we attempt to simplify the first term,

We break apart the number under the square root and find

Simplifying

Therefore we know that in order to try and simplify the other terms, the number under the square root has to be 3. By removing from the other terms in the equation, we will attempt to see if they can be simplified as well.

For the second term,

Finally for the last term,

Our new equation becomes

Once all number under the square roots become the same, we can treat it as simple addition/subtraction and solve.

We then multiply our fraction by because we cannot leave a radical in the denominator. This gives us . Finally, we can simplify our fraction, dividing out a 3, leaving us with

We don't want to have radicals in the denominator. To get rid of radicals, just multiply top and bottom by that radical.

There are two methods we can use to simplify this fraction:

Method 1:

Factor the numerator:

Remember, we need to factor out perfect squares.

Method 2:

You can combine the fraction into one big square root.

Then, you can simplify the fraction.

Let's factor the square roots.

Then, multiply the numerator and the denominator by to get rid of the radical in the denominator.

We can definitely eliminate some answer choices. and don't make sense because we have an irrational number. Next, let's multiply the numerator and denominator of by . When we simplify radical fractions, we try to eliminate radicals, but here, we are going to go backwards.

, so is the answer.

We don't want to have radicals in the denominator. To get rid of radicals, just multiply the numerator and the denominator by that radical.

Remember to distribute the radical in the numerator when multiplying.

This may be the answer; however, the numerator can be simplified. Let's factor out the squares.

Finally, if we factor out a , we get:

Let's get rid of the radicals in the denominator of each individual fraction.

Then find the least common denominator of the fractions, which is , and multiply them so that they each have a denominator of .

We can definitely simplify the numerator in the right fraction by factoring out a perfect square of .

Finally, we can factor out a :

That's the final answer.

To get rid of the radical, we need to multiply by the conjugate. The conjugate uses the opposite sign and multiplying by it will let us rationalize the denominator in this problem. The goal is getting an expression of in which we are taking the differences of two squares.

This answer is the same as . Remember to distribute the negative sign.

To get rid of the radical, we need to multiply by the conjugate. The conjugate uses the opposite sign and multiplying by it will let us rationalize the denominator in this problem. The goal is getting an expression of in which we are taking the differences of two squares.

If we multiplied top and bottom by , we would get nowhere, as this would result: . Instead, let's cross-multiply.

Then, square both sides to get rid of the radical.

Divide both sides by .

The reason the negative is not an answer is because a negative value in a radical is an imaginary number.

Let's combine the two radicals into one radical and simplify.

Remember, when dividing exponents of same base, just subtract the power.

The final answer is .

To get rid of the radical, multiply top and bottom by .

Square both sides.

is "opposite over adjacent," or . (This is because .)

Let's substitute into the equation for and multiply the numerator and the denominator by .

Now we can multiply the result by :

is the inverse of , or :

We can reduce the resulting to :

By multiplying top and bottom by , we can cancel out the in the numerator:

The resulting fraction can be simplified:

At first, this problem seems rather easy. You merely need to divide these two values to get:

However, there are no answers that look like this! When this happens, you should consider rationalizing the denominator to eliminate the square root. This is a little more difficult than normal problems like this (ones that contain only the radical). However, if you complete a difference of squares in the denominator, you will be well on your way to having the right answer:

Carefully FOIL your denominator and distribute your numerator:

Look to your answers for an idea for factoring your numerator:

To find an equivalent, just multiply the top and bottom by the conjugate of the denominator.

Conjugate is the square root expression found in the denominator but with opposite sign.

So:

By simplifying, we get .