Introduction to Exponents

Learn what exponents are and how they simplify repeated multiplication.

Intermediate25 minLesson
Printable worksheet

Definition

An exponent tells you how many times a number is used as a factor in a multiplication.
In the expression 252^5:
  • 22 is the base (the number being multiplied)
  • 55 is the exponent (how many 2s are multiplied together)
25=2×2×2×2×2=322^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
We read 252^5 as "two to the fifth power" or "two to the power of five."
Special names:
  • n2n^2 is called "n squared" (like the area of a square)
  • n3n^3 is called "n cubed" (like the volume of a cube)

Try it now

What does 242^4 mean?

Worked Examples

Calculate 343^4
1

Identify base and exponent

Base = 3, Exponent = 4 → Multiply four 3s together
2

Write out the multiplication

34=3×3×3×33^4 = 3 \times 3 \times 3 \times 3 → Four 3s multiplied
3

Multiply step by step

3×3=93 \times 3 = 9, then 9×3=279 \times 3 = 27, then 27×3=8127 \times 3 = 81

Common Mistakes

Multiplying base by exponent: 34=123^4 = 12

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

Correct: 34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81, not 3×4=123 \times 4 = 12

Swapping base and exponent: treating 262^6 as 6×6=366 \times 6 = 36

Why it's wrong: The base (2) is the number you multiply. The small raised number (6) only counts how many 2s you write.

Correct: 26=2×2×2×2×2×2=642^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64, but 62=6×6=366^2 = 6 \times 6 = 36

Writing the wrong number of zeros: 105=1,000,00010^5 = 1{,}000{,}000

Why it's wrong: Each factor of 10 adds exactly one zero, so the exponent tells you how many zeros to write.

Correct: 105=10×10×10×10×10=100,00010^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100{,}000: a 1 followed by 5 zeros

Interactive Visual

2³ = 8
3

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

No calculations yet

Practice Problems

17 problems
Problem 1 of 17
Easy
What does 242^4 mean?

Why It Matters

Exponents are a powerful shorthand that appears everywhere:
  • Computer storage: Memory is measured in powers of 2 (1 KB = 2102^{10} bytes)
  • Science: Atoms are incredibly small (10−1010^{-10} meters)
  • Finance: Compound interest grows exponentially
  • Biology: Bacteria can double every hour (2242^{24}, more than 16 million in one day!)
Without exponents, writing 101210^{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 = 2102^{10} = 1,024 bytes. 1 Megabyte = 2202^{20} = 1,048,576 bytes.

1Try It Yourself

A small text file is 2122^{12} bytes (4 KB).

How many bytes is that exactly?

Step 1: Write the mathematical expression

Calculate 2122^{12}:

Bacterial Growth

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

Example:

Starting with 1 bacterium, after 10 hours you have 2102^{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 282^8:

Area and Volume

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

Example:

A square with side 7 cm has area 72=497^2 = 49 square cm. A cube with side 4 cm has volume 43=644^3 = 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 535^3:

Key Takeaways

  • 1An exponent tells how many times the base is used as a factor: bn=b×b×...×bb^n = b \times b \times ... \times b (n factors)
  • 2The base is the number being multiplied; the exponent is how many times
  • 3n2n^2 is "n squared" and n3n^3 is "n cubed"
  • 4Powers of 10 are especially useful: 10n10^n is 1 followed by n zeros
  • 5Any number to the power of 1 equals itself: n1=nn^1 = n

Frequently Asked Questions

Any non-zero number to the power of 0 equals 1. For example, 50=15^0 = 1, 1000=1100^0 = 1. This will be covered in detail in a future lesson!
Any non-zero number to the power of 0 equals 1. For example, 50=15^0 = 1, 1000=1100^0 = 1. This will be covered in detail in a future lesson!
'Squared' comes from calculating the area of a square (side times side). 'Cubed' comes from calculating the volume of a cube (side times side times side).
No! 23=82^3 = 8 (three 2s multiplied) but 32=93^2 = 9 (two 3s multiplied). The order matters!

Glossary

Exponent
The small number written above and to the right that shows how many times the base is used as a factor
Base
The number that is multiplied repeatedly in a power
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)
Today's challengeOne problem, three tries

More in This Topic

Laws of Exponents

Intermediate25 min

Square Roots and Cube Roots

Intermediate25 min

Product Rule for Exponents

Intermediate20 min

Quotient Rule for Exponents

Intermediate20 min