What's the difference between 2^3 and 3^2?

2^3 = 2 × 2 × 2 = 8, while 3^2 = 3 × 3 = 9. The base and exponent have different meanings. The base tells you what number to multiply, and the exponent tells you how many times. Switching them gives a different result.

This follows a pattern in exponents. Notice that 5^3 = 125, 5^2 = 25, and 5^1 = 5. Each time the exponent decreases by 1, the result divides by 5. Following this pattern, 5^0 should equal 5 5 = 1. This pattern works for any non-zero base.

Is 2^3 the same as 2 × 3?

No. 2^3 = 8 (meaning 2 × 2 × 2), while 2 × 3 = 6. They are very different. The exponent notation is a compact way to show repeated multiplication of the same number, not multiplication of two different numbers.

How do parentheses change the meaning of an expression with exponents?

Parentheses tell you what to evaluate before applying the exponent. For example, (2 + 3)^2 = 5^2 = 25, but 2 + 3^2 = 2 + 9 = 11. In the first case, you add first, then square the result. In the second case, you square first, then add. Always evaluate what's inside parentheses before applying an exponent outside.

M6MATH-7.1

Numerical Expressions with Exponents

Learn to write and evaluate numerical expressions using whole-number exponents. Understand base and exponent notation, expand powers, and apply the order of operations.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Numerical Expressions with Exponents, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Exponents are a shorthand way to show repeated multiplication. Instead of writing

2 \times 2 \times 2 \times 2

, you can write

2^4

. In this lesson, you'll learn what exponents mean, how to read them, and how to evaluate expressions that contain them. Exponents show up everywhere in math and science—from calculating areas and volumes to describing how bacteria grow. Understanding how to work with them now will make algebra and higher math much easier.

What Is an Exponent?

An exponent is a small number written above and to the right of another number. The number being multiplied is called the base, and the exponent tells you how many times to multiply the base by itself.

For example, in

3^5

, the base is 3 and the exponent is 5. This means

3 \times 3 \times 3 \times 3 \times 3 = 243

. The exponent 5 tells you that 3 is used as a factor 5 times.

It's important to remember that

3^5

does not mean

3 \times 5

. It means 3 multiplied by itself 5 times. This is a place where students often make mistakes—don't multiply the base and exponent together. Instead, use the base as a factor the number of times shown by the exponent.

Exponents can be read in different ways. You can say "3 to the fifth power" or "3 to the fifth." For exponent 2 and 3, there are special names:

5^2

is "5 squared" and

5^3

is "5 cubed." All other exponents are typically read as "to the [number] power."

Evaluating Powers

To evaluate a power means to find its value. Start by writing out the base as a factor as many times as the exponent shows, then multiply.

Let's evaluate

4^3

: This means

4 \times 4 \times 4 = 64

.

Let's evaluate

2^6

: This means

2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

.

One special case: any number to the first power equals itself. So

7^1 = 7

100^1 = 100

, and

x^1 = x

.

Another special case: any non-zero number to the zero power equals 1. So

5^0 = 1

12^0 = 1

, and even

100^0 = 1

. This rule seems strange at first, but it follows from patterns in exponents that you'll explore in later lessons.

When you evaluate a power, be careful to count the factors correctly. Writing out the multiplication helps prevent errors.

Expressions with Exponents and Order of Operations

Many expressions mix exponents with other operations like addition, subtraction, multiplication, and division. The order of operations tells you what to do first: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right), or PEMDAS.

Exponents come right after parentheses in the order, so you evaluate all exponents before you multiply, divide, add, or subtract.

Example: Evaluate

2 + 3^2

. First evaluate

3^2 = 9

. Then add:

2 + 9 = 11

. A common mistake is adding first to get

5^2 = 25

, which is wrong.

Example: Evaluate

5 \times 2^3

. First evaluate

2^3 = 8

. Then multiply:

5 \times 8 = 40

. Don't multiply

5 \times 2

first.

Example: Evaluate

10 - 3^2 + 4

. First evaluate

3^2 = 9

. Then go left to right with subtraction and addition:

10 - 9 + 4 = 1 + 4 = 5

.

When parentheses are involved, evaluate what's inside first. For example,

(2 + 3)^2 = 5^2 = 25

, which is different from

2 + 3^2 = 2 + 9 = 11

Real-World Uses of Exponents

Exponents describe situations where something is multiplied by itself many times. One common example is area and volume. The area of a square with side length 5 units is

5^2 = 25

square units. The volume of a cube with side length 3 units is

3^3 = 27

cubic units.

Exponents also describe exponential growth—when something increases by the same factor repeatedly. If a bacterial colony doubles every hour, after 5 hours the population will be 2 times 2 times 2 times 2 times 2, or

2^5 = 32

times the original size.

Understanding exponents helps you work with very large and very small numbers using scientific notation, a topic you may explore later. It also prepares you for algebra, where exponents are used constantly in formulas and equations.

Key terms

Exponent.: A small number written above and to the right of a base that tells how many times the base is used as a factor.
Base.: The number being multiplied repeatedly in a power.
Power.: A number expressed as a base with an exponent, or the result of evaluating such an expression.
Evaluate.: To find the value of an expression by performing all the operations.
Squared.: Raised to the second power, meaning the base is multiplied by itself once.
Cubed.: Raised to the third power, meaning the base is multiplied by itself twice.
Order of operations.: The agreed-upon rules for which calculations to perform first: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Worked example

Evaluate

2 + 4 \times 3^2 - 5

Step 1: Find all exponents and evaluate them first (order of operations). The only exponent is

3^2

. Evaluate:

3^2 = 3 \times 3 = 9

. The expression becomes

2 + 4 \times 9 - 5

.

Step 2: Next, perform multiplication and division from left to right. The multiplication is

4 \times 9 = 36

. The expression becomes

2 + 36 - 5

.

Step 3: Perform addition and subtraction from left to right. First:

2 + 36 = 38

. Then:

38 - 5 = 33

.

The answer is 33.

Practice questions

Evaluate

5^3

Answer: 125

5^3

means

5 \times 5 \times 5

. First,

5 \times 5 = 25

. Then,

25 \times 5 = 125

. Remember, the exponent 3 tells you to multiply 5 by itself three times, not to multiply 5 by 3.

Which expression equals 16?

$2^4$
$4^2$
$2^3 + 8$
All of the above

Answer: All of the above

2^4 = 2 \times 2 \times 2 \times 2 = 16

4^2 = 4 \times 4 = 16

2^3 + 8 = 8 + 8 = 16

. All three expressions equal 16. This shows that different expressions can have the same value.

Evaluate

3 \times 2^2 + 5

. Show your work.

Answer: 17

Order of operations: First evaluate the exponent

2^2 = 4

. The expression becomes

3 \times 4 + 5

. Next multiply:

3 \times 4 = 12

. Finally add:

12 + 5 = 17

. A common error is multiplying

3 \times 2

first and then squaring, which gives the wrong answer. Always evaluate exponents before you multiply or add.

FAQ

What's the difference between $2^3$ and $3^2$ ?: $2^3 = 2 \times 2 \times 2 = 8$ , while $3^2 = 3 \times 3 = 9$ . The base and exponent have different meanings. The base tells you what number to multiply, and the exponent tells you how many times. Switching them gives a different result.
Why does $5^0 = 1$ ?: This follows a pattern in exponents. Notice that $5^3 = 125$ , $5^2 = 25$ , and $5^1 = 5$ . Each time the exponent decreases by 1, the result divides by 5. Following this pattern, $5^0$ should equal $5 \div 5 = 1$ . This pattern works for any non-zero base.
Is $2^3$ the same as $2 \times 3$ ?: No. $2^3 = 8$ (meaning $2 \times 2 \times 2$ ), while $2 \times 3 = 6$ . They are very different. The exponent notation is a compact way to show repeated multiplication of the same number, not multiplication of two different numbers.
How do parentheses change the meaning of an expression with exponents?: Parentheses tell you what to evaluate before applying the exponent. For example, $(2 + 3)^2 = 5^2 = 25$ , but $2 + 3^2 = 2 + 9 = 11$ . In the first case, you add first, then square the result. In the second case, you square first, then add. Always evaluate what's inside parentheses before applying an exponent outside.

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