Exponent Calculator: How to Calculate Powers, Roots & Scientific Notation
Exponents are everywhere in math, science, finance, and engineering. Whether you need to calculate compound interest, convert between scientific notation and decimal form, or find the n-th root of a number, this guide covers everything you need to know — with clear examples and step-by-step solutions.
What Are Exponents?
An exponent (or power) tells you how many times to multiply a number (the base) by itself. For example, 2⁴ = 2 × 2 × 2 × 2 = 16.
The notation bⁿ means "b raised to the n-th power":
- b = the base (the number being multiplied)
- n = the exponent (how many times to multiply)
Calculating Powers
Positive Integer Exponents
Multiply the base by itself the number of times indicated by the exponent:
3⁵ = 3 × 3 × 3 × 3 × 3 = 243
5³ = 5 × 5 × 5 = 125
2¹⁰ = 1,024
Zero Exponent
Any non-zero number raised to the power of 0 equals 1:
7⁰ = 1
100⁰ = 1
(-3)⁰ = 1
Negative Exponents
A negative exponent means "take the reciprocal and make the exponent positive":
2⁻³ = 1/2³ = 1/8 = 0.125
5⁻² = 1/5² = 1/25 = 0.04
Fractional Exponents
A fractional exponent represents a root:
b^(1/n) = ⁿ√b (the n-th root of b)
8^(1/3) = ³√8 = 2
16^(1/4) = ⁴√16 = 2
27^(2/3) = (³√27)² = 3² = 9
Rules of Exponents
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient Rule | aᵐ / aⁿ = aᵐ⁻ⁿ | 5⁶ / 5² = 5⁴ = 625 |
| Power Rule | (aᵐ)ⁿ = aᵐˣⁿ | (3²)³ = 3⁶ = 729 |
| Power of Product | (ab)ⁿ = aⁿ × bⁿ | (2×3)² = 4 × 9 = 36 |
| Power of Quotient | (a/b)ⁿ = aⁿ / bⁿ | (4/2)³ = 64/8 = 8 |
Calculating Roots
Finding the n-th root of a number is the inverse operation of raising to the n-th power:
- Square root (√): ²√64 = 8, because 8² = 64
- Cube root (³√): ³√27 = 3, because 3³ = 27
- Fourth root (⁴√): ⁴√16 = 2, because 2⁴ = 16
- N-th root: ⁿ√x = x^(1/n)
Note: Even roots of negative numbers (like √(-4)) have no real solution — they produce imaginary numbers. Odd roots of negative numbers do have real solutions: ³√(-27) = -3.
Scientific Notation
Scientific notation expresses very large or very small numbers in a compact form:
a × 10ⁿ where 1 ≤ a < 10
Examples:
6,500,000 = 6.5 × 10⁶
0.000043 = 4.3 × 10⁻⁵
299,792,458 = 2.99792458 × 10⁸ (speed of light in m/s)
Converting to Scientific Notation
- Move the decimal point until you have a number between 1 and 10
- Count how many places you moved the decimal — that is your exponent
- Positive exponent for large numbers (moved left), negative for small (moved right)
Common Scientific Notation Values
| Value | Scientific Notation |
|---|---|
| 1,000 | 1 × 10³ |
| 1,000,000 | 1 × 10⁶ |
| 1,000,000,000 | 1 × 10⁹ |
| 0.001 | 1 × 10⁻³ |
| 0.000001 | 1 × 10⁻⁶ |
Real-World Applications
- Compound interest: A = P(1 + r)ⁿ — uses exponents to calculate growth
- Population growth: Exponential models predict population changes
- Computer science: 2¹⁰ = 1024 (1 kilobyte), 2³² (address space)
- Physics: E = mc² uses an exponent for the speed of light
- Chemistry: pH = -log[H⁺] uses exponents for concentration
- Engineering: Scientific notation for very large or small measurements
Key Takeaways
- bⁿ means multiply b by itself n times
- Zero exponent: any non-zero number to the 0 power equals 1
- Negative exponent: a⁻ⁿ = 1/aⁿ
- Fractional exponent: b^(m/n) = (ⁿ√b)ᵐ
- Learn the five exponent rules: product, quotient, power, product of powers, quotient of powers
- Use our Exponent Calculator for powers, roots, and scientific notation conversions