# GRE Math : Pattern Behaviors in Exponents

## Example Questions

### Example Question #1 : How To Find Compound Interest

A five-year bond is opened with  in it and an interest rate of %, compounded annually. This account is allowed to compound for five years. Which of the following most closely approximates the total amount in the account after that period of time?

Explanation:

Each year, you can calculate your interest by multiplying the principle () by . For one year, this would be:

For two years, it would be:

, which is the same as

Therefore, you can solve for a five year period by doing:

Using your calculator, you can expand the  into a series of multiplications. This gives you , which is closest to

### Example Question #1 : How To Find Compound Interest

Jack has , to invest. If he invests two-thirds of it into a high-yield savings account with an annual interest rate of , compounded quarterly, and the other third in a regular savings account at  simple interest, how much does Jack earn after one year?

Explanation:

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve: Plug in the values given: Therefore, Jack makes$824.32 off his high-yield savings account. Now let's calculate the other interest:

Add the two together, and we see that Jack makes a total of,  off of his investments.

### Example Question #1 : How To Find Compound Interest

If a cash deposit account is opened with  for a three year period at % interest compounded once annually, which of the following is closest to the positive difference between the interest accrued in the third year and the interest accrued in the second year?

Explanation:

It is easiest to break this down into steps. For each year, you will multiply by  to calculate the new value. Therefore, let's make a chart:

After year 1: ; Total interest:

After year 2: ; Let us round this to ; Total interest:

After year 3: ; Let us round this to ; Total interest:

Thus, the positive difference of the interest from the last period and the interest from the first period is:

### Example Question #1 : How To Find Patterns In Exponents

Quantitative Comparison

Quantity A: x2

Quantity B: x3

The two quantities are equal.

Quantity B is greater.

The relationship cannot be determined from the information given.

Quantity A is greater.

The relationship cannot be determined from the information given.

Explanation:

Let's pick numbers. For quantitative comparisons with exponents, it's good to try 0, a negative number, and a fraction.

0: 02 = 0, 03 = 0, so the two quantities are equal.

–1: (–1)2 = 1, (–1)3 = –1, so Quantity A is greater.

### Example Question #261 : Exponents

If , then which of the following must also be true?

Explanation:

We know that the expression must be negative. Therefore one or all of the terms x7, y8 and z10 must be negative; however, even powers always produce positive numbers, so y8 and z10 will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x7 must be negative, so x must be negative. Thus, the answer is x < 0.

### Example Question #1 : How To Find Patterns In Exponents

Which quantity is the greatest?

Quantity A

Quantity B

The realationship cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

Quantity A is greater.

Explanation:

First rewrite quantity B so that it has the same base as quantity A.

can be rewriten as , which is equivalent to .

Now we can compare the two quantities.

is greater than .

### Example Question #4 : How To Find Patterns In Exponents

Simplify the following:

Explanation:

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

Continuing the simplification:

Now, these factors have in common a . Factor this out:

Denominator

This is much simpler:

Cancel out the common factors of :

### Example Question #1 : How To Find Patterns In Exponents

What digit appears in the units place when is multiplied out?

Explanation:

This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.

Observe the first few powers of 2:

2= 2, 2= 4, 2= 8, 2= 16, 2= 32, 2= 64, 2= 128, 2= 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.

The second number in the sequence is 4, so the answer is 4.

### Example Question #21 : Algebra

Which of the following is a multiple of ?

Explanation:

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least  in the prime factorization of the given number.  For each of the answer choices, you have:

### Example Question #1 : How To Find Patterns In Exponents

Simplify the following: