### All SAT Math Resources

## Example Questions

### Example Question #12 : Exponents

If *m* and *n* are integers such that *m* < *n* < 0 and *m*^{2} – *n*^{2} = 7, which of the following can be the value of *m* + *n*?

I. –5

II. –7

III. –9

**Possible Answers:**

I and II only

II and III only

II only

I, II and III only

I only

**Correct answer:**

II only

*m* and *n* are both less than zero and thus negative integers, giving us *m*^{2} and *n*^{2} as perfect squares. The only perfect squares with a difference of 7 is 16 – 9, therefore m = –4 and n = –3.

### Example Question #1 : How To Subtract Exponents

**Possible Answers:**

**Correct answer:**

To simplify, we can rewrite the numerator using a common exponential base.

Now, we can factor out the numerator.

The eights cancel to give us our final answer.

### Example Question #3 : How To Subtract Exponents

Simplify the following expression:

**Possible Answers:**

**Correct answer:**

The correct answer can be found by subtracting exponents that have the same base. Whenever exponents with the same base are divided, you can subtract the exponent of the denominator from the exponent of the numerator as shown below to obtain the final answer:

You do not do anything with the y exponent because it has no identical bases.

### Example Question #1 : How To Subtract Exponents

Solve:

**Possible Answers:**

**Correct answer:**

Subtract the denominator exponent from the numerator's exponent, since they have the same base.

### Example Question #11 : Exponential Operations

Simplify: 3^{2} * (4^{23} - 4^{21})

**Possible Answers:**

3^21

3^3 * 4^21

3^3 * 4^21 * 5

None of the other answers

4^4

**Correct answer:**

3^3 * 4^21 * 5

Begin by noting that the group (4^{23} - 4^{21}) has a common factor, namely 4^{21}. You can treat this like any other constant or variable and factor it out. That would give you: 4^{21}(4^{2} - 1). Therefore, we know that:

3^{2} * (4^{23} - 4^{21}) = 3^{2} * 4^{21}(4^{2} - 1)

Now, 4^{2} - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:

3^{2} * 4^{21}(4^{2} - 1) = 3^{2} * 4^{21}(3 * 5)

Combining multiples withe same base, you get:

3^{3} * 4^{21} * 5

### Example Question #1 : How To Subtract Exponents

Evaluate

**Possible Answers:**

**Correct answer:**

When subtracting exponents, we don't multiply the exponents but we try to factor to see if we simplify the subtraction problem. In this case, we can simplify it by factoring . We get .

### Example Question #6 : How To Subtract Exponents

Evaluate

**Possible Answers:**

**Correct answer:**

Although we have different bases, we know that . As long as exponents are the same, you are allowed to break the base to its prime numbers. Next, we can factor .

### Example Question #7 : How To Subtract Exponents

Evaluate

**Possible Answers:**

**Correct answer:**

Although we have different bases, we know that . Therefore our expression is . Remember to apply the power rule of exponents. Next, we can factor out We get .

### Example Question #8 : How To Subtract Exponents

Simplify so that all exponents are positive:

**Possible Answers:**

**Correct answer:**

When we divide two polynomials with exponents, we subtract their exponents.

Remember that the question asks that all exponents be positive numbers. Therefore:

### Example Question #2 : How To Subtract Exponents

Evaluate:

**Possible Answers:**

**Correct answer:**

When subtracting with exponents, we try to factor out some terms.

We can factor out to get

.