Exponent Calculator
Raise any base to any power — integers, fractions, and negative exponents. Exact BigInt results for large integer powers. Shows expanded steps and scientific notation automatically.
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How It Works
For integer base and whole-number exponent (n ≤ 1000), the calculator uses JavaScript BigInt for exact results — no floating-point rounding. Enter base 2 and exponent 64 for IPv6 address space. The expanded multiplication shows for small values (e.g. 2³ = 2 × 2 × 2). Results above 25 digits show in scientific notation alongside the full exact integer.
aⁿ = a × a × ... × a (n times) | BigInt for exact integers2^10 = 1024 | 2^32 = 4294967296 | 10^9 = 1000000000Enter a negative exponent directly: base 10, exponent −3 → 0.001. The steps panel shows: "Negative exponent: 10^−3 = 1/10³". Negative exponents always give positive results — they represent fractions, not negative numbers. The quick-pick "2^−1" shows 0.5 as a fast demonstration.
a^(−n) = 1/aⁿ | 10^(−3) = 0.001 | 2^(−1) = 0.510^−3 = 0.001 | 2^−4 = 0.0625 | 5^−2 = 0.04Fractional exponents compute roots. Enter 0.5 for square root, 0.333 for cube root, 0.25 for fourth root. The step shows: "Fractional exponent: 8^0.333 = cube root of 8". The quick-pick "2^0.5" gives √2 ≈ 1.41421. Combine: 8^(2/3) = ∛(8²) = ∛64 = 4 — enter base 8, exponent 0.667.
a^(1/n) = ⁿ√a | a^(m/n) = ⁿ√(aᵐ)9^0.5 = 3 | 8^0.333 = 2 | 16^0.25 = 2 | 27^(2/3) = 9Six quick-pick buttons cover the most common cases: 2² (squaring), 3² (perfect square), 10³ (thousands), e² (natural exponential), 2^−1 (half), 2^0.5 (square root). Click any button to prefill both fields instantly, then modify if needed. These clear the result display to avoid confusion with previous calculations.
Quick: 2², 3², 10³, e², 2^−1, 2^0.5Click 10³ → base=10, n=3 → result 1000 instantlyUse exponent rules before calculating to simplify. Product rule: 2³ × 2⁴ = 2^(3+4) = 2^7 = 128. Power rule: (2³)⁴ = 2^(3×4) = 2^12 = 4096. Quotient rule: 2⁶/2² = 2^(6−2) = 2^4 = 16. These reduce multi-step problems to a single exponent calculation.
aᵐ·aⁿ = a^(m+n) | (aᵐ)ⁿ = a^(mn) | aᵐ/aⁿ = a^(m−n)(2³)⁴ = 2^12 = 4096 | 2⁶/2² = 2^4 = 16Quick Reference
Common powers — verify these in the calculator above.
Integer
2^10
1024
Integer
10^6
1,000,000
BigInt
2^32
4,294,967,296
Zero exp
5^0
1
Negative
2^(−3)
0.125
Negative
10^(−6)
0.000001
Root (½)
9^0.5
3
Root (⅓)
8^0.333
≈ 2
Tips & Shortcuts
Use the quick-pick buttons (2², 3², 10³, e², 2^−1, 2^0.5) to prefill common values instantly — then just press Calculate without retyping.
For any nth root, enter 1÷n as the exponent: cube root = exponent 0.333, 5th root = 0.2, 10th root = 0.1. This is faster than switching to another calculator.
The scientific notation result always appears alongside the regular result for large numbers — useful for copying into formulas that expect exponential form.
For very large powers (2^1000 has 302 digits), the full integer result shows with its digit count. This exact output is useful in cryptography and number theory.
To verify an exponent result, switch to the Log/Ln tab: log_b(aⁿ) = n. Enter your result and the base — you should get back your original exponent n.
Common Mistakes
Expecting a negative base with a negative exponent to give a negative result
(−2)^(−3) = 1/(−2)³ = 1/(−8) = −0.125. The sign of the result depends on both the base sign and the exponent: negative base to an odd positive power is negative; to an even positive power is positive; to a negative power follows the same rules then takes the reciprocal.
Entering 2^3^2 expecting left-to-right evaluation
Exponentiation is right-associative: 2^3^2 = 2^(3²) = 2^9 = 512, not (2^3)^2 = 64. If you want left-to-right, calculate (2^3) first (result: 8), then use that as the base with exponent 2.
Using 0.33 for cube root instead of 0.333...
For exact cube root, use 1÷3 ≈ 0.333333. Using 0.33 gives 8^0.33 ≈ 1.987, not exactly 2. For perfect cube roots (8, 27, 64, 125), the tiny error is visible. Use more decimal places or switch to the Root Calculator for exact integer roots.
Confusing aⁿ × aⁿ with a^(n²)
aⁿ × aⁿ = a^(2n) by the product rule — add the exponents. 2³ × 2³ = 2^6 = 64. But (2³)² = 2^(3×2) = 2^6 = 64 too — same result here, but for different exponents: 2³ × 2⁴ = 2^7 = 128 ≠ (2³)^4 = 2^12 = 4096.
Treating a⁰ = 0
a⁰ = 1 for any non-zero a. This follows from the quotient rule: aⁿ/aⁿ = a^(n−n) = a⁰ = 1. Zero exponent does not mean zero result. The only special case is 0⁰, which is conventionally 1 but technically indeterminate.
Frequently Asked Questions
A negative exponent means reciprocal: a^(−n) = 1/aⁿ. So 2^(−3) = 1/8 = 0.125 and 10^(−6) = 0.000001. The result is always positive — negative exponents never produce negative numbers, just fractions. The calculator shows the step "Negative exponent: 2^−3 = 1/2³" explicitly in the steps panel.
Fractional exponent = root. a^(1/n) = ⁿ√a. So 8^(1/3) = ∛8 = 2 and 16^0.25 = ⁴√16 = 2. General rule: a^(m/n) = ⁿ√(aᵐ). Examples: 9^0.5 = √9 = 3. 27^(1/3) = ∛27 = 3. 32^0.2 = ⁵√32 = 2. Enter decimal equivalents: 0.5 for ½, 0.333 for ⅓, 0.25 for ¼.
2^10 = 1024. Powers of 2 are fundamental in computing: 2^8 = 256 (byte values), 2^10 = 1024 (1 KB), 2^16 = 65,536 (16-bit range), 2^32 = 4,294,967,296 (IPv4 addresses), 2^64 ≈ 1.84 × 10^19. The calculator computes these exactly using BigInt — no floating-point rounding.
For integer bases and non-negative integer exponents up to 1000, JavaScript BigInt computes the exact result. 2^100 = 1267650600228229401496703205376 (31 digits, exact). 10^50 has 51 digits — all correct. The result displays with the digit count. Beyond n=1000, standard floating-point is used and the result shows in scientific notation.
Product: aᵐ × aⁿ = a^(m+n) — same base, add exponents. Quotient: aᵐ / aⁿ = a^(m−n) — same base, subtract. Power of power: (aᵐ)ⁿ = a^(mn). Zero exponent: a⁰ = 1 for any a≠0. Negative: a^(−n) = 1/aⁿ. Fractional: a^(1/n) = ⁿ√a. These rules let you simplify expressions before computing.
e² ≈ 7.38906. The constant e ≈ 2.71828 is the base of the natural logarithm. Use the quick-pick button "e²" to compute instantly. Powers of e appear in continuous growth (A = Pe^(rt)), normal distribution (e^(−x²/2)), and Euler's identity (e^(iπ) + 1 = 0). For eˣ with any x, the Antilog tab with base e is purpose-built.
0⁰ is indeterminate in formal mathematics — different contexts assign different values. By the most common convention (used in combinatorics, binomial theorem, and most calculators), 0⁰ = 1. This is because the empty product equals 1, and x⁰ = 1 for all non-zero x. The calculator returns 1, consistent with standard computing behavior (Python, JavaScript, Wolfram Alpha all return 1).
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