Calculate exponentiation and powers with any base and exponent. Enter values into any two fields to solve for the third. Supports scientific notation and the mathematical constant e.
Exponentiation is a mathematical operation written as an, involving the base a and an exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times. This fundamental operation is essential in mathematics, science, engineering, and many real-world applications.
Understanding exponents is crucial for algebra, calculus, scientific calculations, and compound interest computations. Exponents help express very large or very small numbers efficiently and are fundamental to exponential growth and decay models.
Where: a = base, n = exponent
| Base | Exponent | Expression | Result | Description |
|---|---|---|---|---|
| 2 | 3 | 2³ | 8 | 2 × 2 × 2 |
| 5 | 2 | 5² | 25 | 5 × 5 |
| 10 | 4 | 10⁴ | 10,000 | Scientific notation base |
| 3 | 0 | 3⁰ | 1 | Any number to power 0 |
| 4 | -2 | 4⁻² | 0.0625 | 1/4² = 1/16 |
| e | 1 | e¹ | 2.718... | Natural exponential |
| 2 | 10 | 2¹⁰ | 1,024 | Computer science (bytes) |
Euler's Number (e): Approximately 2.71828, e is a fundamental mathematical constant that appears naturally in many areas of mathematics, particularly in calculus and exponential functions.
Natural Exponential Function: ex is the natural exponential function, which has the unique property that its derivative is equal to itself. This makes it incredibly important in calculus and differential equations.
Applications of e: Used in compound interest calculations, population growth models, radioactive decay, and many natural phenomena that exhibit exponential behavior.
Compound Interest: A = P(1 + r/n)nt, where the exponent nt determines how money grows over time with compound interest.
Population Growth: P(t) = P₀ert, where exponential functions model how populations grow over time under ideal conditions.
Computer Science: Powers of 2 are fundamental in computing (28 = 256 for 8-bit systems, 232 for 32-bit addressing).
Scientific Notation: Large numbers are expressed as a × 10n (like 3.0 × 108 m/s for the speed of light).
Fractional Exponents: a1/n represents the nth root of a. For example, 81/3 = ∛8 = 2.
Decimal Exponents: Can be calculated using logarithms or approximation methods. For example, 21.5 = 23/2 = 2√2 ≈ 2.828.
Negative Bases: When the base is negative, the result depends on whether the exponent is even or odd. Even exponents yield positive results, odd exponents yield negative results.
Large Numbers: Use scientific notation for very large results. For example, 106 = 1,000,000 = 1.0 × 106.
Mental Math: Memorize common powers like 2n for n = 1 to 10, and powers of 10 for quick calculations.
Calculator Precision: For very large exponents, results may exceed calculator precision. Consider using logarithms or specialized software for extreme calculations.