What Are Exponents?
An exponent tells you how many times to multiply a number (called the base) by itself. This process is called exponentiation and is written as aⁿ, where:
a is the base
n is the exponent
If n is a positive whole number, the calculation is:
aⁿ = a × a × a × … × a (n times)
This calculator supports negative bases and decimal (fractional) exponents, but does not support imaginary numbers or fractional bases directly.
Key Rules of Exponents
1. Multiplying Same Bases: Add the Exponents
aⁿ × aᵐ = aⁿ⁺ᵐ
Example:
2² × 2⁴ = 2⁶ = 64
2. Negative Exponents: Flip the Base
a⁻ⁿ = 1 / aⁿ
Example:
2⁻³ = 1 / (2 × 2 × 2) = 1/8
3. Dividing Same Bases: Subtract the Exponents
aᵐ / aⁿ = aᵐ⁻ⁿ
Example:
2² / 2⁴ = 2⁻² = 1 / 2² = 1/4
4. Power Raised to Another Power: Multiply the Exponents
(aᵐ)ⁿ = aᵐⁿ
Example:
(2²)⁴ = 2⁸ = 256
5. Multiply Different Bases Raised to Same Exponent
(a × b)ⁿ = aⁿ × bⁿ
Example:
(2 × 4)² = 2² × 4² = 4 × 16 = 64
6. Divide Different Bases Raised to Same Exponent
(a / b)ⁿ = aⁿ / bⁿ
Example:
(2/5)² = 2² / 5² = 4 / 25 = 4/25
7. Exponent of 1
a¹ = a
The number stays the same.
8. Exponent of 0
a⁰ = 1 (except when a = 0)
Why?
Because aⁿ × a⁰ = aⁿ → a⁰ must equal 1 to preserve the rule.
Fractional Exponents (Roots)
A fractional exponent means you’re taking the root of a number.
Tip: Use decimals (e.g., 0.5 instead of 1/2) when entering fractional exponents in the calculator.
Exponents with Negative Bases
This calculator handles negative bases, but only for real number results:
Fractional exponents with negative bases may involve imaginary numbers (e.g., √-1), which are not supported by this tool. You’ll see “NaN” for those results.
