# GRE Math : Square Roots and Operations

## Example Questions

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### Example Question #1 : Square Roots And Operations

Simplify:

Explanation:

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

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

### Example Question #1 : Square Roots And Operations

Compare the quantities.

Quantity A:

Quantity B:

The relationship cannot be determined from the information given.

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

The two quantities are equal.

Explanation:

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!

### Example Question #4 : How To Add Square Roots

Simplify the following expression:

Explanation:

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 :

### Example Question #1 : Square Roots And Operations

Solve for .

Note, :

Explanation:

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 :

### Example Question #1 : Square Roots And Operations

Simplify the following expression:

Explanation:

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.

### Example Question #1 : How To Divide Square Roots

Which of the following is equal to

Explanation:

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

### Example Question #2 : How To Divide Square Roots

Simplify:

Explanation:

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

### Example Question #3 : How To Divide Square Roots

Solve for :

Explanation:

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.

### Example Question #4 : How To Divide Square Roots

Rationalize the denominator:

Explanation:

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

### Example Question #5 : How To Divide Square Roots

Simplify:

Explanation:

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.

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