# SAT Math : Pattern Behaviors in Exponents

## Example Questions

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### 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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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?

Possible Answers:

6

-9

3

-2

-4

Correct answer:

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?

Possible Answers:

p

2p

(2/3)p

3p

(3/2)p

Correct answer:

(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.

Possible Answers:

9

729

27

125

3

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

Explanation:

We use two properties of logarithms:

So

### Example Question #571 : Algebra

Evaluate:

Possible Answers:

Correct answer:

Explanation:

, here  and , hence .

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