OneTaskBy Tim King · Updated
Exponent Calculator
Raise Any Number to a Power, With Exact Big Results
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How the Exponent Calculator works
This exponent calculator raises any base to any power and shows the result in full. Whole-number powers up to 20,000 are computed exactly, so 2^100 shows all 31 digits instead of a rounded figure. Negative exponents also appear as a fraction, fractional exponents are worked as roots, and very large answers come with scientific notation and a digit count.
An exponent says how many times to multiply the base by itself: 2^5 = 2 × 2 × 2 × 2 × 2 = 32. A negative exponent means the reciprocal, so 2^-3 = 1/8. A fractional exponent is a root, so 9^0.5 = 3. Powers grow fast, which is why compound interest, population growth and computer memory sizes are all written with exponents, and why ordinary calculators switch to E notation so quickly.
How the exponent calculator works
The formula is b^n = b × b × … × b, with n copies of b. For whole-number exponents the calculator multiplies exactly using big integers. b^-n = 1 / b^n, and b^(p/q) is the qth root of b^p. Any non-zero base to the power 0 is 1. The digit count comes from the integer part of n × log10(b), plus one.
First example. 2^10 = 1024, which is why a kibibyte is 1,024 bytes.
Second example. 2^-3 = 1 / 2³ = 1/8 = 0.125.
How many digits does 2^100 have? 100 × log10(2) = 30.1, so it has 31 digits: 1,267,650,600,228,229,401,496,703,205,376.
Powers of 2, 3 and 10 from 0 to 24
| n | 2^n | 3^n | 10^n |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
| 1 | 2 | 3 | 10 |
| 2 | 4 | 9 | 100 |
| 3 | 8 | 27 | 1,000 |
| 4 | 16 | 81 | 10,000 |
| 5 | 32 | 243 | 100,000 |
| 6 | 64 | 729 | 1,000,000 |
| 7 | 128 | 2,187 | 10,000,000 |
| 8 | 256 | 6,561 | 100,000,000 |
| 9 | 512 | 19,683 | 1,000,000,000 |
| 10 | 1,024 | 59,049 | 10,000,000,000 |
| 11 | 2,048 | 177,147 | 100,000,000,000 |
| 12 | 4,096 | 531,441 | 1,000,000,000,000 |
| 13 | 8,192 | 1,594,323 | 1e13 |
| 14 | 16,384 | 4,782,969 | 1e14 |
| 15 | 32,768 | 14,348,907 | 1e15 |
| 16 | 65,536 | 43,046,721 | 1e16 |
| 17 | 131,072 | 129,140,163 | 1e17 |
| 18 | 262,144 | 387,420,489 | 1e18 |
| 19 | 524,288 | 1,162,261,467 | 1e19 |
| 20 | 1,048,576 | 3,486,784,401 | 1e20 |
| 21 | 2,097,152 | 10,460,353,203 | 1e21 |
| 22 | 4,194,304 | 31,381,059,609 | 1e22 |
| 23 | 8,388,608 | 94,143,178,827 | 1e23 |
| 24 | 16,777,216 | 282,429,536,481 | 1e24 |
Tips and pitfalls
- Brackets around negatives. -3^2 is -9 by convention; (-3)^2 is 9. The base box here is always the whole base.
- Negative bases need whole exponents. (-8)^(1/3) has a real cube root of -2 on paper, but a decimal like 0.333 is not exactly 1/3, so the calculator asks for a whole-number exponent.
- Zero to the zero. 0^0 is usually taken as 1 in algebra and programming.
- Powers are not multiplication. 3^4 is 81, not 12; the exponent repeats the base.
Sources: Exponentiation.
Checked against: 2^10 = 1,024
FAQ
How do you calculate an exponent?
Multiply the base by itself as many times as the exponent says. 5^3 = 5 × 5 × 5 = 125. For large powers, square repeatedly: 3^8 is 3² = 9, 9² = 81, 81² = 6,561. The calculator does this exactly for whole-number exponents.
What does a negative exponent mean?
A negative exponent means one over the positive power. So 10^-2 = 1/10² = 1/100 = 0.01 and 4^-1 = 1/4. Negative exponents are how scientific notation writes small numbers, such as 3 × 10^-6 for three millionths of a unit.
What is any number to the power of 0?
Any non-zero number to the power 0 is 1. The pattern shows why: 2³ = 8, 2² = 4, 2¹ = 2, and each step divides by 2, so 2⁰ = 1. Zero to the power zero is a special case, usually defined as 1 in algebra.
How do fractional exponents work?
The denominator is a root and the numerator is a power. So 8^(2/3) is the cube root of 8, which is 2, squared to give 4. An exponent of 0.5 is the square root. Enter decimals like 0.5 or 1.5 directly in the exponent box.
How big can the result be?
Whole-number powers up to an exponent of 20,000 are computed exactly. Up to 60 digits are shown in full; longer results show the first 30 and last 20 digits with the total digit count. Beyond that, or for decimal exponents, the calculator gives the result in scientific notation with about 15 significant figures of precision.
See also: all Arithmetic tools, Number Theory, Algebra and Geometry, Statistics, Trigonometry and Calculus.
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