### All ACT Math Resources

## Example Questions

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

For which of the following values of *n* will (**–**3)* ^{n}* represent a real number between 0 and 1?

**Possible Answers:**

1

0

2

**–**2

**–**1

**Correct answer:**

**–**2

In order to transform a negative integer to a positive one, it must be taken to an even power, and to make it a fraction between 0 and 1, it must be taken to a negative power. The only choice that fits both criteria is -2.

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

A professional football player’s contract states that he will earn a salary of $1 million his first year. He would then have a 15% increase every year thereafter for the next 5 years. What would he make in his 6^{th} and final season on the contract?

**Possible Answers:**

**Correct answer:**

We can represent this as an exponential equation (just use million as a label and not a variable):

$1*m ** (1.15)^{5} = $2.01*m*

### Example Question #3 : How To Find Patterns In Exponents

What is the value of the expression *,* given that ?

**Possible Answers:**

**Correct answer:**

First, calculate the value of *m* by taking the two-thirds root of both *m ^{3/2}* and

*64*, leaving

*m=16.*Then solve the expression by plugging in

*16*for

*m*. This gives

*16*

^{2}+ 3(16) = 256 + 48 = 304.* *

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

Evaluate ^{ }

**Possible Answers:**

**Correct answer:**

^{ }

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

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

**Possible Answers:**

**Correct answer:**

We know that the expression must be negative. Therefore one or all of the terms x^{7}, y^{8} and z^{10} must be negative; however, even powers always produce positive numbers, so y^{8} and z^{10} 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, x^{7} must be negative, so x must be negative. Thus, the answer is x < 0.

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

Simplify the following:

**Possible Answers:**

**Correct answer:**

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:

Now, return to your fraction:

Cancel out the common factors of :

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

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

**Possible Answers:**

**Correct answer:**

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^{1 }= 2, 2^{2 }= 4, 2^{3 }= 8, 2^{4 }= 16, 2^{5 }= 32, 2^{6 }= 64, 2^{7 }= 128, 2^{8 }= 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 #1 : How To Find Patterns In Exponents

Which of the following is a multiple of ?

**Possible Answers:**

**Correct answer:**

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:

; This is the answer.

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

Simplify the following:

**Possible Answers:**

**Correct answer:**

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the into its prime factors:

Note that these have a common factor of . Therefore, you can rewrite the numerator as:

Now, put this back into your fraction:

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