Exponent
An exponent (also called a power or index) indicates how many times a base number is multiplied by itself. Written as aⁿ, where a is the base and n is the exponent, it represents repeated multiplication in a compact form. Exponents are essential in scientific notation, polynomial expressions, and exponential growth models.
Key Formula
a^n = a × a × … × a (n times)
LaTeX: a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}
| Symbol | Meaning | Unit |
|---|---|---|
| a | Base number | dimensionless |
| n | Exponent (power) | dimensionless |
Worked Example
Problem
Simplify (2³ × 2⁴) ÷ 2².
Solution
Step 1: Apply product rule: 2³ × 2⁴ = 2^(3+4) = 2⁷ = 128. Step 2: Apply quotient rule: 2⁷ ÷ 2² = 2^(7−2) = 2⁵ = 32.
Answer
2⁵ = 32.
Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
| Power rule | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Interactive Tools
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Related Terms
Logarithm
A logarithm is the inverse operation of exponentiation: log_b(x) = y means b^y = x, answering the question "to what power must b be raised to produce x?" Logarithms transform multiplicative relationships into additive ones, making calculations with very large or small numbers tractable. They are central to information theory, signal processing, and scientific measurement scales such as pH, decibels, and the Richter scale.
Exponential Function
An exponential function is a mathematical function of the form f(x) = aˣ, where the variable x appears as the exponent and the base a is a positive constant not equal to 1. These functions exhibit rapid growth (when a > 1) or decay (when 0 < a < 1) and are fundamental models for population growth, compound interest, radioactive decay, and many natural phenomena.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
From Latin "exponere" (to set forth, to explain). The notation aⁿ was popularised by René Descartes in his 1637 work "La Géométrie", though superscript notation was used informally before then.