Exponent Calculator
Calculate base raised to a power with support for positive, negative, and fractional exponents. Essential for algebra, physics, engineering, and scientific calculations.
Calculate Exponent
Calculate base raised to a power:
b^e = result
Where b is the base and e is the exponent
Example: 2^3 = 8, 10^-2 = 0.01
Understanding Exponents
Exponents represent repeated multiplication and are fundamental to algebra, physics, engineering, and many scientific disciplines. They describe exponential growth, decay, and power relationships.
Exponent Fundamentals
🔢 What is an Exponent?
b^e means b multiplied by itself e times
Read as "b to the power of e" or "b raised to e"
Base × Base × Base ... (e times)
Examples: 2³ = 8, 10² = 100
📊 Special Exponents
Any number^0 = 1 (except 0^0 undefined)
Any number^1 = itself
Negative exponents: b^(-e) = 1/b^e
Fractional exponents: b^(1/n) = nth root
🔬 Scientific Notation
10^n format for very large/small numbers
Mantissa × 10^exponent
Example: 6.02 × 10^23 (Avogadro's number)
Essential for scientific calculations
Exponent Properties
➕ Multiplication Rules
b^m × b^n = b^(m+n)
Same base: add exponents
Example: 2³ × 2⁴ = 2^(3+4) = 2⁷ = 128
Product rule for exponents
➗ Division Rules
b^m ÷ b^n = b^(m-n)
Same base: subtract exponents
Example: 8² ÷ 8⁴ = 8^(2-4) = 8^(-2) = 1/64
Quotient rule for exponents
⚡ Power Rules
(b^m)^n = b^(m×n)
Power to a power: multiply exponents
(b×c)^n = b^n × c^n
Power of a product: distribute exponent
b^0 = 1 (any b ≠ 0)
Practical Applications
💰 Finance & Economics
Compound interest: A = P(1+r)^t
Investment growth calculations
Inflation and depreciation
Economic modeling
Financial forecasting
⚡ Physics & Engineering
Newton's law of universal gravitation
Coulomb's law (electric force)
Radioactive decay equations
Heat transfer calculations
Wave amplitude relationships
🧬 Biology & Chemistry
Population growth models
pH calculations (10^(-pH))
Enzyme kinetics
Drug concentration decay
Bacterial growth curves
💻 Computer Science
Algorithm complexity analysis
Binary calculations (2^n)
Cryptography and hashing
Memory addressing
Data compression
Common Exponent Examples
| Expression | Calculation | Result | Application |
|---|---|---|---|
| 2² | 2 × 2 | 4 | Area calculation |
| 2³ | 2 × 2 × 2 | 8 | Volume calculation |
| 10² | 10 × 10 | 100 | Scientific notation |
| 10^(-2) | 1/(10²) | 0.01 | Centimeters to meters |
| 2^(10) | 2 × 2 × ... × 2 (10 times) | 1024 | Computer memory |
| e^(x) | Euler's number raised to x | 2.718^(x) | Growth/decay models |
🔢 Exponent Tip: Remember that any number raised to the power of 0 equals 1 (except 0^0 which is undefined). Negative exponents mean "one over" the positive power, and fractional exponents represent roots.
Advanced Exponent Concepts
📈 Exponential Functions
f(x) = b^x growth/decay patterns
Natural exponential: e^x
Logarithms as inverse functions
Exponential vs linear growth
Doubling time calculations
🔍 Logarithmic Relationships
If b^e = c, then log_b(c) = e
Common logs: log₁₀(x)
Natural logs: ln(x)
Change of base formula
Logarithmic scales
🌊 Complex Exponents
Euler's formula: e^(iθ) = cosθ + i sinθ
Complex number exponentiation
Polar form of complex numbers
Frequency domain analysis
Quantum mechanics applications
📊 Statistical Applications
Normal distribution calculations
Standard deviation formulas
Correlation coefficients
Regression analysis
Probability distributions
Historical Development
🏛️ Ancient Mathematics
Babylonian base-60 system
Egyptian unit fractions
Greek geometric progressions
Indian mathematical texts
Persian algebraic methods
📚 Renaissance Mathematics
Rafal Bombelli and complex numbers
Francois Viète's algebraic notation
John Napier's logarithms
Isaac Newton's calculus
Leonhard Euler's e and π
⚙️ Modern Applications
Computer algorithm analysis
Financial modeling
Scientific computing
Data encryption
Machine learning algorithms