The Quantitative Reasoning section of the GRE will include several questions testing your understanding of and ability to manipulate exponents or solve exponential equations. Your ability to answer these questions quickly and confidently could make a huge difference in your Quant score. So, in this blog, we are giving you lots of factual information about exponents and plenty of exponent practice problems with answers. The topic is covered in full in our Target Test Prep GRE self-study course.

**Here are the topics we’ll cover:**

Let’s begin by reviewing some basic facts about exponents.

## Basic Facts about Exponents

If you want to perform repeated multiplication of the same number, such as 3 x 3 x 3 x 3 x 3, it is cumbersome to write the entire operation. We use a mathematical shorthand to simplify the instruction. For example, the above can be re-expressed in exponential notation as 3^5, which is pronounced as “3 to the 5th power.” For the exponential expression 3^5, we call the number 3 the base, and the number 5 the exponent. Similarly, if we have the exponential expression 29^8, the base is 29, and the exponent is 8.

KEY FACT:

Repeated multiplication of the same number can be expressed using exponential notation. For example, 7 x 7 x 7 x 7 x 7 x 7 can be expressed as 7^6, where 7 is the base and 6 is the exponent.

### Example 1: Basic Exponents

Which of the following is equivalent to b^5?

- 5b
- b x b x b x b x b
- 5 x 5 x 5 x 5 x 5

- I only
- II only
- III only
- I and II only
- I, II, and III

#### Solution:

The exponential expression b^5 has a base and exponent of b and 5, respectively. This means that we multiply the base b by itself 5 times. Thus, b^5 = b x b x b x b x b.

**Answer: B**

### Negative Exponents

Another basic fact about exponents regards negative exponents, such as 3^(-1), 7^(-2), or 5^(-3). In this situation, the negative exponent indicates that you should move the exponential expression to the denominator of a fraction and eliminate the negative sign from the exponent. Consider these examples:

3^(-1) = 1/ 3^1 = 1 / 3

7^(-2) = 1 / 7^2 = 1 / 49

5^(-3) = 1 / 5^3 = 1 / 125

KEY FACT:

A negative exponent indicates that you should move the exponential expression to the denominator of a fraction and eliminate the negative sign from the exponent.

#### Example 2: Negative Exponents

Which of the following is equivalent to 8 x 2^(-3)?

##### Solution:

We recall that an exponential expression with a negative exponent, like 2^(-3), can be re-expressed as 1 / 2^3, which is equal to 1 / 8. Thus, the expression in the question stem can be re-expressed as:

8 x 2^(-3) = 8 x 1/8 = 1

**Answer: D**

While we have concentrated our discussion here on an exponential expression in the numerator of a fraction, do note that the opposite situation also holds. If the expression with a negative exponent is in the denominator of a fraction, then we must move the exponential expression to the numerator of the fraction and change the sign of the exponent. For example, if we have 1 / 3^(-2), we can move the exponential expression to the numerator and change the negative sign to a positive sign. Thus, we would have:

1 / 3^(-2) = 3^2 = 9

Let’s now consider the common exponent rules.

## Exponent Rules

The exponent rules can be confusing. So, it’s important to practice them until they become second nature to you! Let’s work on the basic exponent rules, one at a time, and get some serious practice with each. The first one is about equal bases.

### Exponent Rule 1: Equal Bases

The rule about equal bases can be illustrated by noting that 2^6 = 2^6, which is pretty obvious. We can take this one step further by observing that if 3^8 = 3^x, then x must equal 8. This isn’t rocket science, but it is a foundational rule for exponential equations.

However, this rule has a few glitches. There are some values that the base can’t take on. For example, the base cannot equal 1. Observe that 1^5 = 1^9 = 1^34 and so forth. In other words, if we are given that 1^6 = 1^x, we might be misled to think that x must equal 6. But we see that when the base is equal to 1, the exponent can be any real number. Thus, in this example, x could be 12 or 8 or even -2.

A similar situation exists for a base of (-1) or a base of 0. In each case, we see that we could have multiple correct answers for the exponent x. For example, if we know that (-1)^3 = (-1)^x, x could be any odd positive integer, since (-1)^3 = (-1)^5 = (-1)^7 = -1. And for a base of 0, we see that no matter what power 0 is raised to (except 0), we always get 0 as the answer. So we see that 0^3 = 0^4 = 0^23 = 0^114 = 0. Thus, we have to carefully write the rule for exponents with equal bases to include these exceptions.

KEY FACT:

In general, when bases are equal, the exponents are equal. For example, if 5^3 = 5^m, then we know that m must equal 3. This rule works for any base except -1, 0, and 1.

Let’s try a Quantitative Comparison question.

#### Example 3: Equal Bases

m^y = m^z and m > 0

- Quantity A is greater than Quantity B
- Quantity B is greater than Quantity A
- Quantities A and B are equal
- The relationship between the two quantities cannot be determined.

##### Solution:

The rule of equal bases can only be used if the base is not equal to -1, 0, or 1. In the question stem, we are told that the base m is greater than 0, but this allows for m to be equal to 1. Thus, we cannot determine a consistent relationship between the two quantities y and z.

**Answer: D**

### Exponent Rule 2: Multiplication of Like Bases

The rule of multiplying like bases can be illustrated by considering the following example:

If we multiply 4^3 and 4^5, we can write it out as (4 x 4 x 4) x (4 x 4 x 4 x 4 x 4). Counting the number of times we multiplied 4 by itself, we see that we have 8 factors of 4. Thus, we could thus express the answer as 4^8. So, a short way of expressing this multiplication would be 4^3 x 4^5 = 4^8. In other words, when we multiply two exponential expressions where the base is the same, we add the exponents. Note that exponents can be positive, negative, or zero.

KEY FACT:

When we multiply exponential expressions with like bases, we add the exponents and don’t change the bases. In general, (a^x) x (a^y) = a^(x + y).

#### Example 4: Multiplying Like Bases

Given: f and g are integers. If 17^f x 17^g = 17^9, then f x g could be any of the following except:

##### Solution:

We use the rule for multiplying exponential expressions with like bases. Since we have the same base of 17, we can add the two exponents, obtaining:

17^f x 17^g = 17^9

17^(f + g) = 17^9

f + g = 9

We are given no restrictions for the values of f and g, but since all the answer choices are positive, let’s consider only nonnegative integer values for f and g:

0 + 9 = 9 , so the product f x g = 0 x 9 = 0

1 + 8 = 9 , so the product f x g = 1 x 8 = 8

2 + 7 = 9 , so the product f x g = 2 x 7 = 14

3 + 6 = 9 , so the product f x g = 2 x 6 = 18

4 + 5 = 9 , so the product f x g = 4 x 5 = 20

**Answer: B**

### Exponent Rule 3: Division of Like Bases

To introduce this rule, let’s consider the problem 4^5 / 4^2. A very basic way of solving this is to expand each of these expressions in the fraction:

4 x 4 x 4 x 4 x 4 / 4 x 4

We see that two of the 4s in the denominator cancel with two of the 4s in the numerator. Thus, the final answer will be:

4^ 5 / 4^2 = 4 x 4 x 4 x 4 x 4 / 4 x 4 = 4 x 4 x 4 = 4^3

This example illustrates the basic concept of division of like bases, which is to subtract the exponent in the denominator from the exponent in the numerator. In other words, if we have 4^5 / 4^2 we can use the technique for dividing exponents to obtain 4^(5 – 2) = 4^3.

KEY FACT:

When we divide exponential expressions with like bases, we subtract the exponent in the denominator from the exponent in the numerator and don’t change the bases. In general, (a^x) / (a^y) = a^(x – y).

#### Example 5: Dividing Like Bases

If x^5 / x^3 = x^a / x^10, then what is the value of a?

##### Solution:

We first use the rule for dividing exponential expressions:

x^5 / x^3 = x^a / x^10

x^(5-3) = x^(a – 10)

x^2 = x^(a – 10)

We now use the rule of equal bases to solve for a:

2 = a – 10

12 = a

**Answer: E**

### Exponent Rule 4: The Power to a Power Rule

Recall that an exponent tells you how many times to multiply the base by itself. So, what if the base is itself an exponential expression? In this case, we would have two distinct powers. For example, if we have (2^4)^3, the base is 2^4, and since we raise that base of 2^4 to the third power, we are to multiply the base 2^4 by itself three times. Thus we have:

(2^4)^3 = 2^4 x 2^4 x 2^4

Now we can use the rule for multiplying like bases to obtain:

(2^4)^3 = 2^4 x 2^4 x 2^4 = 2^(4 + 4 + 4) = 2^12

In essence, we are multiplying exponents to simplify the expression. This technique is far too unwieldy, so we instead can use the power to a power rule to obtain the same result.

KEY FACT:

The power to a power rule states that when we have an exponential expression raised to an exponent, we can multiply the two exponents. For example, (2^4)^3 = 2^(4 x 3) = 2^12.

Let’s try an example.

#### Example 6: The Power to a Power Rule

If a > 1 and (a^-2)^4 = (a^4)^(3x + 4), what is the value of x?

##### Solution:

First, we’ll use the power to a power rule and then the rule of equal bases to solve for x.

(a^-2)^4 = (a^4)^(3x + 4)

a^(-8) = a^4(3x + 4)

a^-8 = a^(12x + 16)

-8 = 12x + 16

-24 = 12x

-2 = x

**Answer: B**

### Exponent Rule 5: When the Two Bases are Not the Same

If we are asked to solve the problem 49^4 = 7^x for the value of x, we see that there is a problem. The bases are not equal.

In situations such as this, the best strategy is to see if there is a way to make the bases equal. In the given example, 49 can be re-expressed as 7^2, so we could re-express the left side with a base of 7 and then use the power to a power rule to solve for x.

49^4 = 7^x

(7^2)^4 = 7^x

7^(2 x 4) = 7^x

7^8 = 7^x

8 = x

KEY FACT:

When the two bases are not the same, try to make them the same by noting an equivalent mathematical relationship between the two bases.

#### Example 7: When the Two Bases Are Not the Same

Which of the following is equal to 3^14? Select all that apply.

##### Solution:

Let’s use the rule for unlike bases to compare each answer choice to the exponential expression in the question stem.

**Choice A**

3^2 x 9^6 = 3^2 x (3^2)^6 = 3^2 x 3^12 = 3^14

Choice A is correct.

**Choice B**

27^3 x 9^4 = (3^3)^3 x (3^2)^4 = 3^9 x 3^8 = 3^17

Choice B is not correct.

**Choice C**

3^5 x 9^3 x 27 = 3^5 x (3^2)^3 x 3^3 = 3^5 x 3^6 x 3^3 = 3^14

Choice C is correct.

**Answer: A, C**

## Summary

Getting a lot of practice with exponents is important for every serious GRE student. In this article, we have provided you with exponent rules practice, including simplifying exponents practice and properties of exponents practice. Let’s review what we’ve covered here:

- Exponential notation is an efficient way of expressing repeated multiplication.
- When we see an exponential expression with a negative exponent in the numerator of a fraction, we move the expression to the denominator and change the negative exponent to a positive one.
- In general, when bases are equal, the exponents are equal. For example, if 5^3 = 5^m, then we know that m must equal 3. This rule works for any base except -1, 0, and 1.
- When we multiply exponential expressions with like bases, we add the exponents and don’t change the bases. In general, (a^x) x (a^y) = a^(x + y).
- When we divide exponential expressions with like bases, we subtract the exponent in the denominator from the exponent in the numerator and don’t change the bases. In general, (a^x) / (a^y) = a^(x – y).
- The power to a power rule states that when we have an exponential expression raised to an exponent, we can multiply the two exponents. For example, (2^4)^3 = 2^(4 x 3) = 2^12.
- When the two bases are not the same, try to make them the same by noting an equivalence mathematical relationship between the two bases.

## What’s Next?

The study of exponents is only one of the many math topics that you will encounter on the GRE Quantitative Reasoning section. In order to see how this topic aligns with the other algebra and math topics, read our article that gives you a GRE Quantitative section overview.