Advanced Exponent Calculator
Calculate powers, roots, and exponential expressions with ease. Perfect for students, engineers, and math enthusiasts worldwide.
Exponent Calculator
Enter any two values to calculate the third:
Understanding Exponents
Exponentiation is a mathematical operation written as aⁿ, where 'a' is the base and 'n' is the exponent. When 'n' is a positive integer, it represents repeated multiplication of the base 'n' times:
aⁿ = a × a × ... × a (n times)
Our calculator handles:
- Positive and negative bases
- Integer and fractional exponents (entered as decimals)
- Negative exponents
- Natural exponential (e)
Any number to power 0: a⁰ = 1 (including 0⁰ = 1)
Any number to power 1: a¹ = a
Negative exponents: a⁻ⁿ = 1/aⁿ
Fractional exponents: a^(1/n) = nth root of a
Exponent Rules and Properties
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aⁿ × aᵐ = aⁿ⁺ᵐ | 2³ × 2² = 2⁵ = 32 |
| Quotient Rule | aⁿ ÷ aᵐ = aⁿ⁻ᵐ | 5⁴ ÷ 5² = 5² = 25 |
| Power Rule | (aⁿ)ᵐ = aⁿᵐ | (3²)³ = 3⁶ = 729 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 4⁻² = 1/16 = 0.0625 |
| Zero Exponent | a⁰ = 1 | 7⁰ = 1 |
| Fractional Exponent | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = (∛8)² = 2² = 4 |
| Product to Power | (ab)ⁿ = aⁿbⁿ | (2×3)² = 2²×3² = 4×9 = 36 |
| Quotient to Power | (a/b)ⁿ = aⁿ/bⁿ | (3/4)² = 3²/4² = 9/16 |
Practical Examples
Formula: A = P(1 + r/n)^(nt)
Example: $1000 invested at 5% annual interest compounded monthly for 3 years:
1000 × (1 + 0.05/12)^(12×3) ≈ $1161.47
Try it: Base = 1.0041667, Exponent = 36
Formula: P = P₀ × e^(rt)
Example: Population starting at 1000 growing at 2% per year for 10 years:
1000 × e^(0.02×10) ≈ 1221.40
Try it: Check "Use e", Exponent = 0.2
3.5 × 10⁴ = 35,000
2.1 × 10⁻³ = 0.0021
Try it: Base = 10, Exponent = 4 or -3
√25 = 25^(1/2) = 5
∛8 = 8^(1/3) = 2
Try it: Base = 25, Exponent = 0.5
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