What is an Exponent?

Exponentiation is a mathematical operation involving a base (a) and an exponent (n), written as aⁿ. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base n times:

aⁿ = a × a × ... × a (n times)

This concept is commonly used in mathematics, especially in algebra and beyond. The calculator provided on this page can handle negative bases and fractional exponents (as decimals), but it does not compute imaginary numbers.

Basic Exponent Laws and Rules

Multiplication of Exponents (Same Base)

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

aⁿ × aᵐ = a⁽ⁿ⁺ᵐ⁾

Example: 2² × 2⁴ = 4 × 16 = 64
2² × 2⁴ = 2^(2 + 4) = 2⁶ = 64

Negative Exponents

A negative exponent flips the base into the denominator and changes the exponent to positive:

a⁻ⁿ = 1 / aⁿ

Example: 2⁻³ = 1 / 2³ = 1/8

Division of Exponents (Same Base)

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

aᵐ / aⁿ = a⁽ᵐ⁻ⁿ⁾

Example: 2² / 2⁴ = 4 / 16 = 1/4
2² / 2⁴ = 2^(2-4) = 2⁻² = 1/4

Power of a Power

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

(aᵐ)ⁿ = a⁽ᵐˣⁿ⁾

Example: (2²)⁴ = 2^(2×4) = 2⁸ = 256

Multiplying Bases Raised to the Same Exponent

When multiplying bases with the same exponent, distribute the exponent:

(a × b)ⁿ = aⁿ × bⁿ

Example: (2 × 4)² = 8² = 64
2² × 4² = 4 × 16 = 64

Dividing Bases Raised to the Same Exponent

When dividing bases with the same exponent, distribute the exponent:

(a / b)ⁿ = aⁿ / bⁿ

Example: (2/5)² = 2² / 5² = 4 / 25

Exponent of 1

Any base to the power of 1 is itself:

a¹ = a

Exponent of 0

Any non-zero base raised to the power of 0 is always 1:

a⁰ = 1

**Proof:** If aⁿ × a⁰ = aⁿ, then a⁰ must be 1 for the equation to hold true.

Fractional Exponents

A fractional exponent represents a root. For example:

a^(1/n) = the nth root of a

a^(m/n) = (nth root of a) raised to the m

Note: The calculator accepts fractional exponents if entered as decimals (e.g., 0.5 instead of 1/2).

Negative Bases with Exponents

Negative bases follow similar rules but affect the sign depending on whether the exponent is even or odd:

• Even exponent: Result is positive.
• Odd exponent: Result is negative.

Fractional exponents with negative bases typically require imaginary numbers, which this calculator does not support. Inputs resulting in imaginary numbers will return “NaN” (Not a Number).

Summary

This guide explains the fundamental rules of exponents and provides practical examples for better understanding. These rules are essential in algebra, calculus, and many areas of science and engineering.

Use our Exponent Calculator above to quickly solve exponent problems with ease!
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