Powers grow astonishingly fast. 2¹⁰ is a modest 1,024, but 2¹⁰⁰ already has 31 digits and 2¹⁰⁰⁰ has 302. Most calculators give up and switch to rounded scientific notation long before that. This one keeps every digit: it uses exact big-integer arithmetic, so you can see the complete value of numbers like 2¹⁰⁰⁰ or 123456789⁵⁰⁰⁰, count their digits, and read their first and last digits.
How to use the large exponents calculator
- Enter the base, a whole number of any length. Negative numbers are allowed.
- Enter the exponent, a whole number of 0 or more.
- The tape shows the number of digits, scientific notation, the first and last 20 digits, the trailing zeros and the digit sum. The complete number is printed in the scrollable box below the steps.
How big powers are computed
The value is defined as repeated multiplication:
Multiplying n times would be slow for large n, so the calculator uses exponentiation by squaring. Write the exponent in binary, square the base repeatedly (b, b², b⁴, b⁸, …) and multiply together only the squares that match a 1 bit. For n = 1,000 (binary 1111101000) that is 14 multiplications instead of 999.
The number of digits comes from logarithms, which is also how the calculator decides whether a result fits within its limit:
Worked example: 2¹⁰⁰⁰
Size: 1,000 × log₁₀ 2 ≈ 1,000 × 0.30103 = 301.03, so 2¹⁰⁰⁰ has ⌊301.03⌋ + 1 = 302 digits.
Scientific notation: 10^0.03 ≈ 1.0715, so 2¹⁰⁰⁰ ≈ 1.0715 × 10³⁰¹.
Exact value: it begins 10,715,086,071,862,673,209… and ends …24,386,837,205,668,069,376. Its digits add up to 1,366.
Notice that 2¹⁰⁰⁰ ends in 6. Powers of 2 cycle through the last digits 2, 4, 8, 6, and 1,000 is a multiple of 4, so the last digit is 6. Patterns like this are a quick way to check a long result.
Reading the extra results
Digit count and scientific notation
The digit count answers “how big is it?” more honestly than the scientific notation’s leading digits. For comparison, the number of atoms in the observable universe is often estimated at around 10⁸⁰, a number with 81 digits, while 2³⁰⁰ already has 91 digits.
Trailing zeros
A power ends in zeros only if its base does. 10ⁿ has n trailing zeros, and 20⁵ = 3,200,000 has 5. Every factor of 10 needs a 2 and a 5, so a base like 2 or 7 never produces a trailing zero.
Digit sum
The digit sum is a classic checksum. A number is divisible by 3 exactly when its digit sum is, and by 9 when its digit sum is. Since 2 is not divisible by 3, no power of 2 is either, and 2¹⁰⁰⁰’s digit sum of 1,366 is indeed not a multiple of 3.
Where huge exact powers matter
- Cryptography. RSA encryption raises numbers with hundreds of digits to large powers (modulo another number) every time you connect to a secure website.
- Combinatorics. The number of possible 20-character passwords from 94 printable characters is 94²⁰, a 40-digit number.
- Number theory. The largest known primes are Mersenne primes of the form 2ᵖ − 1 with tens of millions of digits.
For decimal or fractional exponents, use the exponent calculator. Factorials grow even faster than powers; compare with the factorial calculator.
Frequently asked questions
Why does a regular calculator show 2^1000 as 1.07E+301?
Ordinary calculators and spreadsheets store numbers as floating point with about 15 to 17 significant digits, so they round huge results and show scientific notation. This calculator uses arbitrary-precision integer arithmetic, so all 302 digits of 2^1000 are exact.
How many digits does b^n have?
For a positive whole number base, the digit count is ⌊n·log₁₀ b⌋ + 1. For 2^1000 that is ⌊1000 × 0.30103⌋ + 1 = 302.
Is there a limit?
Results are capped at 100,000 digits so the page stays responsive. That still covers 2^332,190, 10^99,999 and 99^50,000. For bigger powers the exponent calculator gives the value in scientific notation.
Can the base be negative?
Yes. A negative base gives a negative result when the exponent is odd and a positive result when it is even, because each pair of negative factors multiplies to a positive.