Exponent Calculator
Calculate any base raised to any power. Supports positive, negative, and fractional exponents. Shows expanded form and fraction conversion.
Exponents
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Result
25 = 32
Exponential Curve f(x) = 2x
Step-by-Step Solution
Power Definition
25 = 2 raised to the 5th power
Expanded Form
2 × 2 × 2 × 2 × 2 = 32
Inverse Root
5th root of 2 = 1.148698355
Exponent Rules
bᵐ × bⁿ
= bm+n
bᵐ ÷ bⁿ
= bm−n
(bᵐ)ⁿ
= bmn
b−n
= 1/bⁿ
The Formula
Exponentiation indicates repeated multiplication. A positive integer exponent means multiply the base by itself that many times. A negative exponent means take the reciprocal. A fractional exponent represents a root. The exponent can be any real number, including fractions, decimals, negatives, and irrational numbers.
Variable Definitions
Base
The value being raised to a power. Can be positive, negative, or decimal.
Exponent
The power to raise the base to. Can be any real number: integer, fraction, decimal, or negative.
Power
The result of raising b to the nth power. The fundamental operation of exponentiation.
How to Use This Calculator
- 1
Enter the base value (positive, negative, or decimal).
- 2
Enter the exponent (integer, fractional, negative, or decimal).
- 3
View the result, expanded form (for small integer exponents), and scientific notation for very large or small results.
- 4
Check the inverse (root) — the calculator shows the corresponding root of the base.
- 5
Check the reciprocal (1/result) — useful for understanding the relationship between a number and its multiplicative inverse.
Exponentiation is repeated multiplication: the base multiplied by itself exponent times
Understanding the Concept
Exponents are one of the most fundamental operations in mathematics. They appear everywhere — from compound interest (A = P(1+r)ⁿ) to scientific notation (6.022 × 10²³) to geometric growth (population doubling). The laws of exponents (product rule: bᵐ × bⁿ = bᵐ⁺ⁿ, quotient rule: bᵐ ÷ bⁿ = bᵐ⁻ⁿ, power rule: (bᵐ)ⁿ = bᵐⁿ) make it possible to simplify complex expressions. When the exponent is a fraction like 1/2, it represents the square root: b^(1/2) = √b. A negative exponent means reciprocal: b⁻ⁿ = 1/bⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. Zero exponent: any non-zero number raised to the power of 0 equals 1 (b⁰ = 1). The calculator shows expanded form for small integer exponents, displaying the repeated multiplication explicitly. For very large results (like 2¹⁰⁰), scientific notation is used to keep the output readable. Exponentiation is not commutative: 2³ = 8 but 3² = 9 — the order of base and exponent matters.
Frequently Asked Questions
Sources & References
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