# PSAT Math : Pattern Behaviors in Exponents

## Example Questions

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### Example Question #1 : Pattern Behaviors In Exponents

On January 15, 2015, Philip deposited $10,000 in a certificate of deposit that returned interest at an annual rate of 8.125%, compounded monthly. How much will his certificate of deposit be worth on January 15, 2020? Possible Answers: Correct answer: Explanation: The formula for compound interest is where is the initial investment, is the interest rate expressed as the decimal equivalent, is the number of periods per year the interest is compounded, is the number of years, and is the final value of the investment. Set (monthly = 12 periods), and , and evaluate : The CD will be worth$14,991.24.

### Example Question #2 : Pattern Behaviors In Exponents

Money is deposited in corporate bonds which yield 6.735% annual interest compounded monthly, and which mature after ten years. Which of the following responses comes closest to the percent by which the value of bonds increases?

Explanation:

The formula for compound interest is

where  is the initial investment,  is the interest rate expressed as the decimal equivalent,  is the number of periods per year the interest is compounded,  is the number of years, and  is the final value of the investment.

In the given scenario, , and  (monthly); substitute:

This meas that the final value of the bonds is about 1.96 times their initial value, or, equivalently, 96% greater than their initial value. Of the given responses, 95% comes closest.

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

Donna wants to deposit money into a certificate of deposit so that in exactly ten years, her investment will be worth $100,000. The interest rate of the CD is 7.885%, compounded monthly. What should Donna's initial investment be, at minimum? Possible Answers: More information is needed to answer the question. Correct answer: Explanation: The formula for compound interest is where is the initial investment, is the interest rate expressed as the decimal equivalent, is the number of periods per year the interest is compounded, is the number of years, and is the final value of the investment. Set (monthly = 12 periods), and , and evaluate : The correct response is$45,569.99.

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

Tom invests , in a savings account with an annual interest rate of . If his investment is compounded semiannually, how much interest does he earn after  years?

Explanation:

In order to find the interest earned, used the compound interest formula

where  represents the number of times the account is compounded each year, and  represents the interest rate expressed as a decimal.

The account is worth $16882.63 after two years. Therefore Tom earns$1882.63 in interest.

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

If ax·a4 = a12 and (by)3 = b15, what is the value of x - y?

3

-4

-2

6

-9

3

Explanation:

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.

### Example Question #1 : Pattern Behaviors In Exponents

If p and q are positive integrers and 27= 9q, then what is the value of q in terms of p?

2p

(2/3)p

3p

p

(3/2)p

(3/2)p

Explanation:

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 33p = 32q. So then 3p = 2q, and q = (3/2)p is our answer.

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

Simplify 272/3.

3

27

125

729

9

9

Explanation:

272/3 is 27 squared and cube-rooted. We want to pick the easier operation first. Here that is the cube root. To see that, try both operations.

272/3 = (272)1/3 = 7291/3 OR

272/3 = (271/3)2 = 32

Obviously 32 is much easier. Either 32 or 7291/3 will give us the correct answer of 9, but with 32 it is readily apparent.

### Example Question #5 : Pattern Behaviors In Exponents

If  and  are integers and

what is the value of ?

Explanation:

To solve this problem, we will have to take the log of both sides to bring down our exponents. By doing this, we will get .

To solve for  we will have to divide both sides of our equation by  to get .

will give you the answer of –3.

### Example Question #6 : Pattern Behaviors In Exponents

If and , then what is ?

Explanation:

We use two properties of logarithms:

So

Evaluate: