Exponent Calculator
Raise any base number to any exponent — positive, negative, or fractional. The calculator handles integer powers like 2¹⁰ = 1024, fractional exponents like 8^(1/3) = 2, and negative exponents like 2^(−3) = 0.125. The content below explains exponent rules, scientific notation, and real-world applications.
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Formula
bⁿ = b × b × b × … (n times)
Exponentiation is repeated multiplication. b is the base (the number being multiplied), n is the exponent (how many times you multiply). For positive integer n: bⁿ = b × b × … × b (n factors). For negative n: b⁻ⁿ = 1/bⁿ. For fractional n = p/q: b^(p/q) = the qth root of b^p. Zero exponent: b⁰ = 1 for any nonzero b.
How to use the Exponent Calculator
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Laws of exponents
Six rules govern all exponent arithmetic. Knowing them lets you simplify complex expressions without a calculator.
1. Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ. Multiplying same-base powers: add exponents. 2³ × 2⁵ = 2⁸ = 256. 2. Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Dividing same-base powers: subtract exponents. 3. Power rule: (aᵐ)ⁿ = aᵐⁿ. A power raised to a power: multiply exponents. (2³)² = 2⁶ = 64. 4. Zero exponent: a⁰ = 1 for a ≠ 0. 5. Negative exponent: a⁻ⁿ = 1/aⁿ. 2⁻³ = 1/8 = 0.125. 6. Fractional exponent: a^(m/n) = ⁿ√(aᵐ). 8^(2/3) = (³√8)² = 2² = 4.
Scientific notation and very large numbers
Scientific notation expresses numbers as a × 10ⁿ where 1 ≤ a < 10. It's indispensable when exponents create numbers too large or too small for standard notation.
2¹⁰ = 1,024 ≈ 10³ (that's why a 'kilobyte' is 1024 bytes, not 1000 — computing uses powers of 2, not 10). 2³² = 4,294,967,296 ≈ 4.3 × 10⁹. This is the number of addresses in IPv4 — and why the internet ran out of them.
Avogadro's number (6.022 × 10²³) and the number of atoms in a grain of sand (~10¹⁸) both rely on exponents to be expressible at all. Exponents make the scale of the universe — from quarks to galaxy clusters, a range of roughly 10⁴² — tractable to write.
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Exponential growth and compound interest
Exponential functions (where the variable is in the exponent) describe growth that accelerates over time. Compound interest follows A = P × (1 + r)ⁿ — your balance after n periods is the principal times a power of (1 + r).
Population growth, viral spread, radioactive decay (with a negative exponent), and Moore's Law all follow exponential models. A key property: doubling time is constant. A 7% annual growth rate doubles in about 10 years (the rule of 72: 72 ÷ 7 ≈ 10).
Exponential growth is often underestimated early and then shocks people later. The classic lily pad puzzle: if a lily doubles every day and covers the pond on day 30, it was only half-covered on day 29.
Tips & Insights
Negative base with even/odd exponent
A negative base raised to an even exponent is positive: (−2)⁴ = 16. Raised to an odd exponent, it's negative: (−2)³ = −8. This is because an even number of negative factors pair up to give positive products.
The rule of 72
Divide 72 by a percentage growth rate to estimate doubling time. At 6% annual growth, money doubles in 72/6 = 12 years. At 12%, it doubles in 6 years. This works because ln(2) ≈ 0.693 ≈ 0.72 and ln(1+r) ≈ r for small r.
0⁰ is conventionally defined as 1
The expression 0⁰ is technically indeterminate (the limits of xʸ as x,y→0 depend on the path). But in combinatorics, computer science, and the binomial theorem, 0⁰ = 1 is the useful convention that makes formulas work correctly.
Worked Examples
Computer storage (bytes to terabytes)
2⁴⁰ = 1,099,511,627,776 ≈ 1.1 × 10¹². This is exactly 1 tebibyte (TiB) — 2⁴⁰ bytes. A 1 TB drive advertised in decimal (10¹²) contains slightly fewer bytes than the 2⁴⁰ tebibyte definition.
Compound interest over 30 years
1.07³⁰ ≈ 7.61. Every dollar invested at 7% annual return is worth $7.61 after 30 years — a 661% gain purely from compounding, without adding any additional principal.
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Frequently Asked Questions
What does a negative exponent mean?
A negative exponent means the reciprocal of the positive exponent: b⁻ⁿ = 1/bⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125. Negative exponents are how scientific notation expresses small numbers: 10⁻⁶ = 0.000001 (one millionth).
What does a fractional exponent mean?
A fractional exponent represents a root. b^(1/n) = ⁿ√b (the nth root of b). More generally, b^(m/n) = ⁿ√(bᵐ). So 8^(1/3) = cube root of 8 = 2, and 4^(3/2) = (√4)³ = 2³ = 8.
Why is anything to the power of 0 equal to 1?
Using the quotient rule: aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰. But aⁿ/aⁿ = 1 (any nonzero number divided by itself). Therefore a⁰ = 1. Another way to see it: aⁿ ÷ a = aⁿ⁻¹. Dividing by a repeatedly: a² → a¹ → a⁰ = 1.
What is the difference between 2³ and 3²?
2³ = 2 × 2 × 2 = 8. 3² = 3 × 3 = 9. Exponentiation is not commutative: swapping base and exponent gives different results (unless base = exponent, as in 2² = 4 and still 2² = 4, which is trivial).
How large can this calculator handle?
JavaScript's number type (IEEE 754 double precision) handles values up to about 1.8 × 10³⁰⁸ (Number.MAX_VALUE). Beyond that, results appear as Infinity. For cryptographic or scientific work with very large integers, specialized big-number libraries are needed.
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