Take each fraction separately first:

(2√3)/(√2) = [(2√3)/(√2)] * [(√2)/(√2)] = (2 * √3 * √2)/(√2 * √2) = (2 * √6)/2 = √6

Similarly:

(4√2)/(√3) = [(4√2)/(√3)] * [(√3)/(√3)] = (4√6)/3 = (4/3)√6

Now, add them together:

√6 + (4/3)√6 = (3/3)√6 + (4/3)√6 = (7/3)√6

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 

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 .

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.

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.