Exponent Calculator
Calculate the result of raising a number to any power. Supports positive, negative, and decimal exponents.
Result
Quick Examples
What Are Exponents?
An exponent tells you how many times to multiply a number (the base) by itself. For example, 2³ means 2 × 2 × 2 = 8. The small raised number is called the exponent or power.
bn = b × b × b × ... (n times)
Exponent Rules
Product: am × an = am+n
Quotient: am ÷ an = am-n
Power of Power: (am)n = am×n
Zero Exponent: a0 = 1
Negative Exponent: a-n = 1/an
Powers of 2
| Exponent | Value | Common Name |
|---|---|---|
| 2¹⁰ | 1,024 | 1 KB |
| 2²⁰ | 1,048,576 | 1 MB |
| 2³⁰ | 1,073,741,824 | 1 GB |
Fractional and Decimal Exponents
Exponents don't have to be whole numbers. Fractional exponents are another way to write roots:
Square root: a^(1/2) = √a e.g. 16^(1/2) = √16 = 4
Cube root: a^(1/3) = ∛a e.g. 27^(1/3) = ∛27 = 3
General rule: a^(m/n) = (ⁿ√a)^m e.g. 8^(2/3) = (∛8)² = 2² = 4
Powers of 10: Scientific Notation Reference
Powers of 10 are the backbone of the metric system and scientific notation. Each step up multiplies the value by 10.
| Power | Value | Metric Prefix | Example |
|---|---|---|---|
| 10⁻⁶ | 0.000001 | micro (μ) | 1 μm = 1 micron |
| 10⁻³ | 0.001 | milli (m) | 1 mm = 0.001 m |
| 10⁰ | 1 | — | Base unit |
| 10³ | 1,000 | kilo (k) | 1 km = 1,000 m |
| 10⁶ | 1,000,000 | mega (M) | 1 MHz = 10⁶ Hz |
| 10⁹ | 1,000,000,000 | giga (G) | 1 GB = 10⁹ bytes |
| 10¹² | 1 trillion | tera (T) | 1 TB = 10¹² bytes |
Real-World Applications of Exponents
Compound Interest
Future value = P × (1 + r)^n. A $1,000 investment at 7% annual return for 30 years grows to $1,000 × 1.07^30 ≈ $7,612 — entirely due to the power of exponents.
Computer Science & Storage
Binary (base 2) uses exponents of 2. A 64-bit processor handles 2^64 ≈ 18.4 quintillion values. Storage units (KB, MB, GB, TB) are all powers of 2 or 10.
Population & Bacterial Growth
Bacteria doubling every hour grow by 2^n in n hours. Starting from 1 cell: after 20 hours there are 2^20 = 1,048,576 cells.
Richter Scale & Sound (dB)
These logarithmic scales are the inverse of exponents. A magnitude 7 earthquake releases 10× more energy than magnitude 6. Every +10 dB sound is 10× more intense.
Step-by-Step Examples
Example 1: Compound Interest (Positive Exponent)
$1,000 invested at 7% annual return for 20 years
FV = 1000 × 1.07²⁰
1.07²⁰ = 3.8697
FV = $3,869.68
Example 2: Negative Exponent (Metric Prefix)
1 microsecond = 10⁻⁶ seconds
10⁻⁶ = 1 / 10⁶ = 1 / 1,000,000 = 0.000001
Example 3: Fractional Exponent (Square Root)
√144 = 144^(0.5) = 144^(1/2)
144^(1/2) = 12 (because 12 × 12 = 144)
Complete Exponent Rules Reference
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power of power | (aᵐ)ⁿ = aᵐˣⁿ | (2³)² = 2⁶ = 64 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |
| Power of product | (ab)ⁿ = aⁿ × bⁿ | (2×3)² = 4 × 9 = 36 |
Common Exponent Mistakes
aᵐ × bⁿ ≠ (ab)ᵐ⁺ⁿ
The product rule only applies when the base is the same. 2³ × 3² cannot be simplified to a single power — they have different bases.
Negative exponent ≠ negative result
3⁻² = 1/9, not −9. A negative exponent means reciprocal, not sign change.
(a + b)ⁿ ≠ aⁿ + bⁿ
This is one of the most common algebra errors. (2 + 3)² = 25, but 2² + 3² = 13. The power must be distributed using the binomial theorem, not term by term.