Power Function Calculator

Calculate any number raised to any power with step-by-step solutions and detailed explanations

Calculate Power Function: f(x) = b^x

The number to be raised to a power

The power to which the base is raised

Power Function Results

Error
Invalid: 0 raised to zero or negative power is undefined

Example Calculations

Positive Integer Exponent

Example:

Calculation: 2 × 2 × 2 = 8

Result: 8

Negative Exponent

Example: 3⁻²

Calculation: 1/(3²) = 1/9

Result: 0.111111...

Zero Exponent

Example: 5⁰

Rule: Any number⁰ = 1

Result: 1

Decimal Exponent

Example: 4⁰·⁵

Calculation: √4

Result: 2

Power Function Rules

Zero Exponent

a⁰ = 1 (for a ≠ 0)

First Power

a¹ = a

Negative Exponent

a⁻ⁿ = 1/aⁿ

Fractional Exponent

a^(m/n) = ⁿ√(aᵐ)

Product Rule

aᵐ × aⁿ = a^(m+n)

Quotient Rule

aᵐ ÷ aⁿ = a^(m-n)

Quick Presets

Understanding Power Functions

What is a Power Function?

A power function is a mathematical function of the form f(x) = b^x, where b is the base and x is the exponent. The exponent determines how many times the base is multiplied by itself.

Types of Exponents

  • Positive integers: Repeated multiplication (e.g., 2³ = 2×2×2)
  • Zero: Always equals 1 (except 0⁰)
  • Negative: Reciprocal of positive power
  • Fractions: Roots and powers combined

Important Rules

Zero Exponent Rule

Any non-zero number raised to the power of 0 equals 1

a⁰ = 1 (where a ≠ 0)

Negative Exponent Rule

A negative exponent means taking the reciprocal

a⁻ⁿ = 1/aⁿ

Fractional Exponent Rule

A fractional exponent represents roots

a^(1/n) = ⁿ√a

Applications

  • Scientific calculations and measurements
  • Exponential growth and decay models
  • Computer science and algorithms
  • Financial calculations (compound interest)

Special Cases and Notes

Undefined Cases

  • • 0⁰ is mathematically undefined
  • • 0 raised to any negative power is undefined
  • • Division by zero in negative exponents

Large Numbers

  • • Results may be displayed in scientific notation
  • • Very large numbers may exceed calculation limits
  • • Decimal precision may be limited for very small results