# SAT Math : Pattern Behaviors in Exponents

## Example Questions

← Previous 1 3 4

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

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

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 of the interest accrued between the last two years and the first two years?

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 #2 : 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 #11 : New Sat Math No Calculator

A truck was bought for  in 2008, and it depreciates at a rate of  per year. What is the value of the truck in 2016? Round to the nearest cent.

Explanation:

The first step is to convert the depreciation rate into a decimal. . Now lets recall the exponential decay model. , where  is the starting amount of money,  is the annual rate of decay, and  is time (in years). After substituting, we get

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

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

6

-9

3

-2

-4

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 #2 : 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?

p

2p

(2/3)p

3p

(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.

9

729

27

125

3

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 #661 : Algebra

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 #1 : How To Find Patterns In Exponents

If and , then what is ?

Explanation:

We use two properties of logarithms:

So

Evaluate: