Exponent Calculator
Calculate the result of raising a base number to an exponent power. Supports positive, negative, integer, and fractional exponents with scientific notation.
Calculate Exponential Power
Enter a base number and exponent to calculate the result:
Supports negative and fractional exponents
What is an Exponent?
An exponent represents how many times a number (base) is multiplied by itself. For example, 2³ = 2 × 2 × 2 = 8, where 2 is the base and 3 is the exponent.
Mathematical Notation
where b is the base and n is the exponent
Types of Exponents
📈 Positive Integer
Multiplication repeated n times
Example: 2³ = 2 × 2 × 2 = 8
➖ Negative Integer
1 divided by positive power
Example: 2^(-3) = 1/(2³) = 1/8 = 0.125
🔢 Fractional
nth root of the base
Example: 8^(1/3) = ∛8 = 2
0️⃣ Zero Exponent
Any number to power of 0 equals 1
Example: 5⁰ = 1
Laws of Exponents
📊 Product Rule
b^m × b^n = b^(m+n)
Example: 2² × 2³ = 2^(2+3) = 2⁵ = 32
➗ Quotient Rule
b^m ÷ b^n = b^(m-n)
Example: 2⁵ ÷ 2² = 2^(5-2) = 2³ = 8
⚡ Power Rule
(b^m)^n = b^(m×n)
Example: (2³)² = 2^(3×2) = 2⁶ = 64
🔄 Negative Rule
b^(-n) = 1/b^n
Example: 2^(-3) = 1/2³ = 1/8
Scientific Notation
Scientific notation expresses very large or small numbers as a number between 1 and 10 multiplied by a power of 10.
where 1 ≤ a < 10 and b is an integer
Examples:
| Regular Number | Scientific Notation | Explanation |
|---|---|---|
| 1,000,000 | 1 × 10⁶ | 10 to the power of 6 |
| 0.000001 | 1 × 10^(-6) | 10 to the power of -6 |
| 2,500,000 | 2.5 × 10⁶ | 2.5 times 10 to the power of 6 |
| 0.0000003 | 3 × 10^(-7) | 3 times 10 to the power of -7 |
Common Exponent Calculations
| Expression | Result | Description |
|---|---|---|
| 2² | 4 | 2 squared (2 × 2) |
| 2³ | 8 | 2 cubed (2 × 2 × 2) |
| 10² | 100 | 10 squared |
| 10³ | 1,000 | 10 cubed |
| 2^(-1) | 0.5 | 1 divided by 2 |
| 4^(1/2) | 2 | Square root of 4 |
| 8^(1/3) | 2 | Cube root of 8 |
Applications of Exponents
🏗️ Engineering
Stress calculations, signal processing, exponential growth models
💰 Finance
Compound interest, inflation calculations, investment growth
🔬 Science
Radioactive decay, population growth, chemical reaction rates
📊 Data Science
Exponential smoothing, statistical distributions
💡 Pro Tip: When dealing with very large or small numbers, scientific notation makes calculations and comparisons much easier. Remember that any number raised to the power of 0 equals 1, and negative exponents represent fractions!
Special Cases
- Zero as Base: 0^n = 0 for n > 0, undefined for n = 0, undefined for n < 0
- One as Base: 1^n = 1 for any n
- One as Exponent: b¹ = b for any b
- Zero as Exponent: b⁰ = 1 for any b ≠ 0