What Is an Exponent?

An exponent is a mathematical notation used to represent repeated multiplication of a number, called the base. Exponentiation is written as aⁿ, where a is the base and n is the exponent. When the exponent is a positive integer, it shows how many times the base is multiplied by itself:

aⁿ = a × a × ... × a  (n times)

For example:

2³ = 2 × 2 × 2 = 8

The calculator above can handle negative bases and fractional exponents (in decimal form) but cannot compute imaginary numbers.

Basic Exponent Rules

Understanding the basic rules of exponents makes calculations simpler.

1. Multiplying Exponents with the Same Base

When multiplying numbers with the same base, add the exponents:

a^m × aⁿ = a^m+n

Example:

2² × 2⁴ = 2²⁺⁴ = 2⁶ = 64

2. Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent:

a^-n = 1 / aⁿ

Example:

2^-3 = 1 / 2³ = 1 / 8

3. Dividing Exponents with the Same Base

When dividing numbers with the same base, subtract the exponents:

a^m / aⁿ = a^m-n

Example:

2² / 2⁴ = 2^2-4 = 2^-2 = 1/4

4. Power of a Power

When raising an exponent to another exponent, multiply the exponents:

(a^m)ⁿ = a^m×n

Example:

(2²)⁴ = 2^2×4 = 2⁸ = 256

5. Power of a Product or Quotient

Product: (a × b)ⁿ = aⁿ × bⁿ
Example: (2 × 4)² = 2² × 4² = 64
Quotient: (a / b)ⁿ = aⁿ / bⁿ
Example: (2 / 5)² = 2² / 5² = 4 / 25

6. Exponent of 1 and 0

Exponent 1: a¹ = a
Exponent 0: a⁰ = 1 (for any non-zero base)

7. Fractional Exponents

A fractional exponent indicates a root of the base:

a^1/n = √n(a)

8. Exponents with Negative Bases

Even positive exponent → result is positive
Odd positive exponent → result is negative
Fractional exponents with negative bases → may result in imaginary numbers (calculator cannot handle these)

Key Takeaways

An exponent represents repeated multiplication of a base.
Positive integers = multiply the base multiple times.
Negative exponent = reciprocal of the base raised to a positive exponent.
Fractional exponent = nth root of the base.
Exponent rules simplify calculations involving multiplication, division, and powers.
Negative bases may produce negative or imaginary results depending on the exponent.