Exponent Calculator
Calculate any base raised to any power — including negative and fractional exponents.
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How Exponents Work
An exponent (or power) indicates how many times a base number is multiplied by itself. The expression xⁿ means "x multiplied by itself n times." Exponents are fundamental in algebra, geometry, science, finance (compound interest), and computer science (binary arithmetic).
Positive Integer Exponents
2³ = 2 × 2 × 2 = 8. The base is 2, the exponent is 3. Any number to the first power is itself (x¹ = x). Any non-zero number to the zero power is 1 (x⁰ = 1). This is because dividing like bases subtracts exponents: x¹/x¹ = x⁰ = 1.
Negative Exponents
A negative exponent means "take the reciprocal": x⁻ⁿ = 1/xⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. Negative exponents are common in scientific notation for very small numbers.
Fractional Exponents
A fractional exponent represents a root: x^(1/n) = ⁿ√x. For example, 8^(1/3) = ∛8 = 2. And x^(m/n) = (ⁿ√x)^m. So 8^(2/3) = (∛8)² = 2² = 4.
Laws of Exponents
Key rules: xᵃ × xᵇ = x^(a+b), xᵃ / xᵇ = x^(a−b), (xᵃ)ᵇ = x^(a×b), (xy)ⁿ = xⁿ × yⁿ.
Worked Example: Applying Exponent Rules
Simplify (2³ × 2⁵) / 2⁴.
Step 1 — Multiply: 2³ × 2⁵ = 2^(3+5) = 2⁸ = 256.
Step 2 — Divide: 2⁸ / 2⁴ = 2^(8−4) = 2⁴ = 16.
So (2³ × 2⁵) / 2⁴ = 16. You could also write it as 8 × 32 / 16 = 256 / 16 = 16 — same answer.
Powers of 2 Reference
| Exponent | 2ⁿ | Common name |
|---|---|---|
| 2⁰ | 1 | One |
| 2⁸ | 256 | Byte values |
| 2¹⁰ | 1,024 | ~1 kilobyte |
| 2²⁰ | 1,048,576 | ~1 megabyte |
| 2³⁰ | 1,073,741,824 | ~1 gigabyte |
Frequently Asked Questions
xⁿ means x multiplied by itself n times. 2⁴ = 2×2×2×2 = 16.
Any non-zero number raised to the power of 0 equals 1. This follows from the exponent subtraction rule: xⁿ/xⁿ = x⁰ = 1.
x⁻ⁿ = 1/xⁿ. For example, 3⁻² = 1/9 ≈ 0.111. It means reciprocal of the positive power.
2¹⁰ = 1,024. Powers of 2 are fundamental in computing — 2¹⁰ is approximately 1 thousand, 2²⁰ is about 1 million.
x^(1/n) = the nth root of x. x^(m/n) = (nth root of x) raised to the m. Example: 27^(2/3) = (∛27)² = 3² = 9.
Because multiplying a negative number by itself twice always gives a positive: (−3)² = (−3) × (−3) = 9. In general, (−x)^n is positive when n is even and negative when n is odd. So (−2)³ = −8, but (−2)⁴ = 16.
2³ = 2 × 2 × 2 = 8, while 3² = 3 × 3 = 9. Exponentiation is not commutative — the base and exponent cannot be swapped freely. In general, aᵇ ≠ bᵃ.