Exponent Calculator
Calculate powers and roots. Compute baseⁿ and nth roots.
What is an exponent?
An exponent (also called a power or index) tells you how many times to multiply a number — the base — by itself. Written as b^n, the base b is multiplied n times. Exponents are fundamental to mathematics, science, computing, and finance, appearing everywhere from compound interest calculations to scientific notation and binary code.
2^3 = 2 × 2 × 2 = 8
5^4 = 5 × 5 × 5 × 5 = 625
10^6 = 1,000,000 (one million)Special exponent rules
| Rule | Formula | Example |
|---|---|---|
| Zero exponent | b⁰ = 1 | 7⁰ = 1, 1000⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent (square root) | b^0.5 = √b | 9^0.5 = 3, 25^0.5 = 5 |
| Fractional exponent (nth root) | b^(1/n) = ⁿ√b | 8^(1/3) = ∛8 = 2 |
| Power of a power | (bᵐ)ⁿ = b^(m×n) | (2³)² = 2⁶ = 64 |
| Product rule | bᵐ × bⁿ = b^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient rule | bᵐ ÷ bⁿ = b^(m−n) | 2⁶ ÷ 2² = 2⁴ = 16 |
Powers of 2: the foundation of computing
Powers of 2 are the backbone of binary arithmetic and digital storage. Every bit doubles capacity, which is why storage sizes jump from 256 MB to 512 MB to 1,024 MB (1 GB) rather than rounding to the nearest 500.
2¹ = 2 2⁸ = 256 2¹⁶ = 65,536
2² = 4 2¹⁰ = 1,024 2²⁰ = 1,048,576 (≈ 1 million)
2³ = 8 2¹²= 4,096 2³² = 4,294,967,296 (≈ 4 billion)
2⁴ = 16 2¹⁴= 16,384 2⁶⁴ = 18,446,744,073,709,551,616Real-world applications of exponents
Exponential growth vs. linear growth
Linear growth adds the same amount each period. Exponential growth multiplies by the same factor each period. The difference is staggering over time — a hallmark of why compound interest, viral content, and technology adoption curves are so powerful.
Linear growth (add 10 each year):
Year 0: 100 | Year 5: 150 | Year 10: 200 | Year 20: 300
Exponential growth (multiply by 1.1 each year = 10% annual growth):
Year 0: 100 | Year 5: 161 | Year 10: 259 | Year 20: 673Frequently asked questions
What does an exponent mean?
An exponent tells you how many times to multiply a number (the base) by itself. For example, 2⁵ means 2 × 2 × 2 × 2 × 2 = 32.
What is a negative exponent?
A negative exponent means the reciprocal: 2⁻³ equals 1 ÷ 2³ = 1/8. The base moves to the denominator.
What does a fractional exponent do?
A fractional exponent is a root. For example, 9^(1/2) is the square root of 9 = 3, and 8^(1/3) is the cube root of 8 = 2.
What is anything to the power of zero?
Any non-zero number raised to the power of 0 equals 1. This keeps the rules of exponents consistent.
xⁿ = x multiplied by itself n times. x^0.5 = square root. x^−1 = 1/x.