Exponents — Mathematics Knowledge System

Overview

An exponent tells you how many times a number (the base) is multiplied by itself. Writing 2⁵ instead of 2×2×2×2×2 is a compact notation that becomes essential once numbers get large, and it obeys its own set of algebraic rules for combining, multiplying, and dividing powers. Exponents extend naturally beyond positive whole numbers to zero, negative, and even fractional values, each with a consistent meaning that preserves the same rules.

Intuition

If you think of multiplication as repeated addition (3×4 means 'add 3 four times'), exponentiation is repeated multiplication: 3⁴ means 'multiply 3 by itself four times.' Each step up — addition, multiplication, exponentiation — is a shorthand for repeating the previous operation. This is also why exponents grow so explosively: 2¹⁰ is already 1024, while 10×2 is only 20. That explosive growth is exactly what makes exponents useful for describing compound interest, population growth, and computer memory sizes.

Interactive Graph

Formal Definition

Definition

For a base a and a positive integer n, aⁿ denotes a multiplied by itself n times. The definition extends to zero and negative integer exponents so that the core exponent rules remain valid for all integers.

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}

Definition (n a positive integer)

a^0 = 1 \quad (a \neq 0)

Zero exponent

a^{-n} = \frac{1}{a^n} \quad (a \neq 0)

Negative exponent

Notation

Notation	Meaning
$a^n$	a raised to the power n; a is the base, n is the exponent
$a^2$	a squared
$a^3$	a cubed

Properties

Product of powers

a^m \cdot a^n = a^{m+n}

Quotient of powers

\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)

Power of a power

(a^m)^n = a^{mn}

Power of a product

(ab)^n = a^n b^n

Applications

Compound interest formulas, A = P(1+r)ᵗ, use exponents to model growth that compounds over repeated periods.

Worked Examples

Multiply 2 by itself 4 times.
$2 \times 2 \times 2 \times 2 = 16$

Answer: 16

Practice Problems

Difficulty 1/10

Evaluate 5³.

Difficulty 2/10

Simplify x⁵ · x² and write the answer as a single power.

Common Mistakes

Common Mistake

Multiplying the base by the exponent instead of repeated multiplication, e.g. thinking 3⁴ = 3×4 = 12.

Exponentiation is repeated MULTIPLICATION, not multiplication by the exponent: 3⁴ = 3×3×3×3 = 81.

Common Mistake

Adding exponents when the bases are different, e.g. writing 2³ · 3² as 6⁵.

The product-of-powers rule aᵐ·aⁿ = aᵐ⁺ⁿ only applies when the bases are the SAME. With different bases, each power must be evaluated separately before combining.

Summary

An exponent aⁿ represents a multiplied by itself n times.
a⁰ = 1 (for a ≠ 0), and a⁻ⁿ = 1/aⁿ, extending exponents beyond positive integers.
Key rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ.
These rules only combine powers directly when the bases match.
Exponential growth (repeated multiplication) increases far faster than linear or multiplicative growth.

References

WebsiteWikipedia — Exponentiation