Introduction to Exponents

Learn what exponents are and how they simplify repeated multiplication.

Intermediate25 minLesson

Definition

An exponent tells you how many times to multiply a number by itself.

In the expression

2^5

$2$ is the base (the number being multiplied)
$5$ is the exponent (how many times to multiply)

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

We read

2^5

as "two to the fifth power" or "two to the power of five."

Special names:

$n^2$ is called "n squared" (like the area of a square)
$n^3$ is called "n cubed" (like the volume of a cube)

Try it now

What does

2^4

mean?

Worked Examples

Calculate $3^4$

Identify base and exponent

Base = 3, Exponent = 4 → Multiply 3 by itself 4 times

Write out the multiplication

$3^4 = 3 \times 3 \times 3 \times 3$ → Four 3s multiplied

Multiply step by step

$3 \times 3 = 9$ , then $9 \times 3 = 27$ , then $27 \times 3 = 81$

Answer: $3^4 = 81$

Common Mistakes

Multiplying base by exponent: $3^4 = 12$

Why it's wrong: This confuses exponentiation with multiplication. The exponent tells you how many times to multiply, not what to add.

Correct: $3^4 = 3 \times 3 \times 3 \times 3 = 81$ , not $3 \times 4 = 12$

Thinking $2^3 = 2^2 + 2^1$

Why it's wrong: Exponents don't work like regular addition. You can't split them this way.

Correct: $2^3 = 8$ , while $2^2 + 2^1 = 4 + 2 = 6$ . These are different!

Writing $5^2$ as $25^1$ and thinking they're different

Why it's wrong: Any number to the power of 1 equals itself: $25^1 = 25$

Correct: $5^2 = 25$ and $25^1 = 25$ are the same value, just written differently

Interactive Visual

Base:

Calculate:2^3 = 8

See how powers of a number grow on the number line. Change the base and exponent.

Factor Tree

Number:

Composite

Prime

Enter a number to see its factor tree and prime factorization.

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

17 problems

Problem 1 of 17

Easy

What does $2^4$ mean?

Why It Matters

Exponents are a powerful shorthand that appears everywhere:

Computer storage: Memory is measured in powers of 2 (1 KB = $2^{10}$ bytes)
Science: Atoms are incredibly small ( $10^{-10}$ meters)
Finance: Compound interest grows exponentially
Biology: Bacteria can double every hour ( $2^{24}$ = 16 million in one day!)

Without exponents, writing

10^{12}

(one trillion) would require 12 zeros!

Real World Applications

Computer Memory

Computer memory is measured in powers of 2 because computers use binary (0s and 1s).

Example:

1 Kilobyte = $2^{10}$ = 1,024 bytes. 1 Megabyte = $2^{20}$ = 1,048,576 bytes.

1Try It Yourself

A small text file is about $2^{12}$ bytes (4 KB).

How many bytes is that exactly?

Step 1: Write the mathematical expression

Calculate $2^{12}$ :

Bacterial Growth

Bacteria multiply by splitting in two. Each hour, the population doubles.

Example:

Starting with 1 bacterium, after 10 hours you have $2^{10}$ = 1,024 bacteria!

2Try It Yourself

You start with 1 bacterium that doubles every hour.

How many bacteria after 8 hours?

Step 1: Write the mathematical expression

Calculate $2^8$ :

Area and Volume

Exponents naturally appear in geometry: area uses squared, volume uses cubed.

Example:

A square with side 7 cm has area $7^2 = 49$ square cm. A cube with side 4 cm has volume $4^3 = 64$ cubic cm.

3Try It Yourself

You have a cube-shaped box with sides of 5 cm.

What is the volume of the box?

Step 1: Write the mathematical expression

Calculate $5^3$ :

Key Takeaways

1An exponent tells how many times to multiply a base by itself: $b^n = b \times b \times ... \times b$ (n times)
2The base is the number being multiplied; the exponent is how many times
3 $n^2$ is "n squared" and $n^3$ is "n cubed"
4Powers of 10 are especially useful: $10^n$ is 1 followed by n zeros
5Any number to the power of 1 equals itself: $n^1 = n$

Frequently Asked Questions

Any non-zero number to the power of 0 equals 1. For example,

5^0 = 1

100^0 = 1

. This will be covered in detail in a future lesson!

Glossary

Exponent: The small number written above and to the right that shows how many times to multiply the base by itself
Base: The number that gets multiplied by itself in exponential notation
Power: The result of raising a base to an exponent (e.g., 8 is a power of 2)
Squared: Raised to the power of 2 (multiplied by itself once)
Cubed: Raised to the power of 3 (multiplied by itself twice)

Introduction to Exponents

Definition

Worked Examples

Common Mistakes

Interactive Visual

Factor Tree

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Computer Memory

Bacterial Growth

Area and Volume

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Laws of Exponents

Square Roots and Cube Roots