Exponent & Power Calculator

Calculate any base raised to positive, negative, zero, or fractional exponents (b^e) with scientific notation, reciprocals (b⁻ᵉ), and natural logarithms.

Category: Math

How to Use Exponent & Power Calculator

  1. Enter the Base Number (b) and the Exponent Power (e) into the input fields (positive, negative, zero, and decimal values are supported).
  2. Click Calculate to compute b^e.
  3. Review the exact power value, scientific notation (m × 10ⁿ), multiplicative reciprocal (b⁻ᵉ = 1 / bᵉ), and natural logarithm ln(bᵉ).

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Frequently Asked Questions

How do negative exponents work (such as 2⁻³)?

A negative exponent represents the multiplicative reciprocal of the positive power: b⁻ᵉ = 1 / (bᵉ). For example, 2⁻³ = 1 / 2³ = 1 / 8 = 0.125. Note that a base of 0 cannot be raised to a negative exponent because 1 / 0ᵉ causes division by zero.

What is 0 raised to the power of 0 (0⁰)?

In algebra, combinatorics, set theory, and the IEEE-754 floating-point standard, 0⁰ is defined as 1 (the empty product). In calculus and mathematical analysis, the limit of x^y as both x and y approach 0 is an indeterminate form whose value depends on the path taken.

How do fractional (decimal) exponents relate to roots?

A fractional exponent b^(m/n) equals the n-th root of b raised to the m-th power: ⁿ√(bᵐ). For example, 16^0.5 = √16 = 4, and 8^(1/3) = ∛8 = 2. For negative bases, non-integer exponents generally yield complex numbers and are flagged by the real-number calculator.

Why does the calculator show Scientific Notation alongside the standard result?

Exponents grow or shrink rapidly (for example, 2⁵⁰ ≈ 1.1259 × 10¹⁵ or 2⁻²⁰ ≈ 9.5367 × 10⁻⁷). Scientific notation expresses any result as m × 10ⁿ (written m e±n), making very large or microscopic powers easy to read and compare.

What is the relationship between exponents and the Natural Logarithm ln(bᵉ)?

Logarithms are the inverse operation of exponentiation. For any positive result bᵉ > 0, the natural logarithm satisfies the power identity ln(bᵉ) = e × ln(b), which is widely used in continuous compounding, physics, and algorithmic complexity.