### All GRE Math Resources

## Example Questions

### Example Question #1 : Exponents And The Distributive Property

What is the value of t if: 3x^{2 } + tx - 21 = (3x - 3)(x + 7)?

**Possible Answers:**

**Correct answer:**18

Use the foil method: (3x - 3) (x + 7) = 3x^{2} +21x - 3x - 21 = 3x^{2} +18x -21 so t = 18.

### Example Question #2 : Exponents And The Distributive Property

Expand the following equation:

**Possible Answers:**

**Correct answer:**

use FOIL to factor the expression.

First: (x^{3})(x) = x^{4}

Outside (x^{3})(7) = 7x^{3}

Inside (–3)(x) = –3x (Don't forget the negatives!)

Last (3)(7) = –21

### Example Question #1 : Exponents And The Distributive Property

Quantitative Comparison

Quantity A:

Quantity B:

**Possible Answers:**

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

**Correct answer:**

Quantity B is greater.

Quantity A: 2^{2} + 3^{2} = 4 + 9 = 13

Quantity B: (2 + 3)^{2} = 5^{2} = 25, so Quantity B is greater.

We can also think of this in more general terms. *x*^{2} + *y*^{2} does not generally equal (*x* + *y*)^{2}.

### Example Question #1 : Exponents

Quantity A:

Quantity B:

**Possible Answers:**

The two quantities are equal.

The relationship cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

**Correct answer:**

Quantity B is greater.

The difference of squares formula says (x + a)(x - a) = x^{2} - a^{2}.

Thus, Quantity A equals 8.

Therefore, Quantity B is greater.

### Example Question #5 : How To Use Foil

Quantity A:

Quantity B:

**Possible Answers:**

Quantity A is greater.

Quantity B is greater.

The relationship cannot be determined.

The two quantities are equal.

**Correct answer:**

Quantity B is greater.

To approach this problem, consider the two quantities

Quantity A:

Quantity B:

They are in different forms, so expand quantity A:

Quantity A:

Quantity B:

Now, for the purpose of comparison, subtract shared terms from each quantity:

Quantity A*:

Quantity B*:

Both and are negative, non-zero values. Since is a product of two negative values, it must be positive. Quantity B must be greater than Quantity A.

### Example Question #6 : How To Use Foil

Expand the function:

**Possible Answers:**

**Correct answer:**

Use the method of FOIL (First, Outside, Inside, Last) and add exponents for like bases:

### Example Question #7 : How To Use Foil

Quantity A:

Quantity B:

**Possible Answers:**

The relationship cannot be determined.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

**Correct answer:**

The relationship cannot be determined.

Begin by expanding Quantity A:

Now in order to compare this to Quantity B:

A good method would be to subtract shared terms from each Quantity; in this case, both quantities have an and term. Removing them gives:

Quantity A' :

Quantity B' :

The question now is the sign of Quantity A'; if it's always positive, Quantity A is greater. If it's always negative, Quantity B is greater. If it is zero, the two are the same.

We only know that

If , then Quantity A' would be zero.

If , then Quantity A' would be positive.

Since values of x and y can be chosen to vary the relationship, th relationship cannot be determined.

### Example Question #2 : Exponents

Quantity A:

Quantity B:

**Possible Answers:**

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined.

**Correct answer:**

Quantity B is greater.

Begin by expanding Quantity A:

Now in order to compare this to Quantity B:

A good method would be to subtract shared terms from each Quantity; in this case, both quantities have an and term. Removing them gives:

Quantity A' :

Quantity B' :

The question now is the sign of Quantity A'; if it's always positive, Quantity A is greater. If it's always negative, Quantity B is greater. If it is zero, the two are the same.

We know that

Now compare and :

Looking at absolute values so that we're only considering positive terms:

From this it follows that by multiplying across the inequality :

From this we can determine that the magnitude of is greater. However, since this is the product of one negative number and two positive numbers, is negative, and the sum of and must in turn be negative, and so Quantity A' must be negative!

From this we can say that Quantity B is greater.

### Example Question #2 : Exponents

Quantity A:

Quantity B:

**Possible Answers:**

The two quantities are equal.

Quantity B is greater.

The relationship cannot be determined.

Quantity A is greater.

**Correct answer:**

The relationship cannot be determined.

This problem is deceptive. Looking at Quantity A, one may think to factor and reduce it as follows:

Which is identical to Quantity B.

However, we cannot ignore that in the original fraction! We are given no conditions as to the value of . If , then Quantity A would be undefined. Since we're not given the condition , we cannot ignore this possibility.

The relationship cannot be established.

### Example Question #3 : Exponents

Solve the following expression, .

**Possible Answers:**

**Correct answer:**

You must FOIL the expression which means to multiply the first terms together followed by the outer terms, then the inner terms and lastly, the last terms.

The expression written out looks like

.

You multiple both First terms to get .

Then the outer terms are multiplied .

Then you multiple the inner terms together .

Finally you multiply the last terms of each .

This gives you or .

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