Exponent Calculator: Powers, Logs and Roots
Raise a number to any power, solve for the exponent or find the n-th root. Handles negative and fractional exponents, exact big integers and scientific notation, free and private in your browser.
Raise the base to an exponent: base^exponent. The exponent may be negative, fractional such as 1/3, or zero.
Enter a base and an exponent to see the result.
About the Exponent Calculator
The Exponent Calculator raises a number to a power and also works backwards. In Power mode you enter a base and an exponent and get the result, including negative, fractional and zero exponents. In Logarithm mode you enter a base and a result and get the exponent that links them. In Root mode you enter an exponent n and a result and get the base, which is the n-th root. Answers update as you type, so there is no button to press.
Whole-number powers are exact. The calculator builds them with arbitrary-precision integers, so 2^100 prints as 1267650600228229401496703205376 rather than a rounded float. Results with more than 60 digits switch to scientific notation with 10 significant digits, correctly rounded from the exact value, so 7^80 shows as 4.053621560e+67 and 10^400 shows as 1.000000000e+400. Every other result is printed to 12 significant digits, which hides binary floating point noise.
How to use the exponent calculator
Steps
- Pick a mode: Power, Logarithm or Root.
- Fill in the two fields for that mode. Power uses Base and Exponent, Logarithm uses Base and Result, and Root uses Exponent (n) and Result.
- Read the answer. Whole exponents up to 12 also show the repeated product, such as 2 x 2 x 2 = 8.
- Press Copy to copy the answer.
Fields accept negative signs, decimals, thousands commas and scientific notation such as 1.5e3. In Power mode the exponent may also be a fraction like 1/3, which is handy for roots.
Formulas
Worked examples
Example 1: a large exact power
Enter base 7 and exponent 80. The exact value has 68 digits, more than the 60 that fit on screen, so the calculator prints 4.053621560e+67. That means 4.053621560 times 10 to the power 67, rounded to 10 significant digits.
Example 2: find the exponent
In Logarithm mode, enter base 10 and result 5000. The answer is 3.69897000434, and the check line confirms that 10 to that power gives 5,000. For results that are an exact power, such as base 2 and result 1024, the answer is the whole number 10 instead of 9.99999999999.
Example 3: find the base
In Root mode, enter exponent n = 5 and result -32. The answer is -2, because (-2) x (-2) x (-2) x (-2) x (-2) = -32. An odd root of a negative number is real. With n = 4 and result -16 there is no real answer, and the calculator says so.
Special cases
- Exponent 0 gives 1 for every non-zero base. Exponent 1 gives the base itself.
- 0^0 is shown as 1 with a visible note: it is indeterminate in mathematics, and 1 is the common convention in algebra and programming.
- 0 to a negative power is an error, because it means dividing by zero.
- A negative base with a fractional exponent has no real result in Power mode. For an odd root such as the cube root of -8, use Root mode.
- Outside the supported range: whole-number powers are exact up to about a million bits. Beyond that, or when a decimal result overflows a double, you get an inline message. A non-zero result too small for a double is shown as 0 with a note that it rounds to 0.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
The Percentage Calculator handles growth and change in percent, the Pascal's Triangle Calculator builds the binomial coefficients that appear when you expand powers of a sum, and the Number List Calculator sums and averages a list of values.
Frequently asked questions
How do I calculate a number to a power?
Choose Power mode, type the base and the exponent, and the result appears as you type. For a whole exponent from 2 to 12 the calculator also shows the repeated product, for example 2 x 2 x 2 = 8. The exponent can be negative, fractional such as 1/3, or zero.
How does the calculator find an exponent or a base?
Logarithm mode solves base^x = result using x = ln(result) / ln(base), and it needs a base above 0 that is not 1 and a result above 0. Root mode solves base^n = result using base = result^(1/n), and n must not be zero. Both show a check line that raises the answer back to the power.
What is 0 to the power 0?
It is indeterminate in mathematics, because the rules for exponents pull it towards different values. Most algebra, combinatorics and programming languages define it as 1, so the calculator shows 1 with a note saying so.
Why does a negative base with a fractional exponent give an error?
A power such as (-8)^(1/3) is ambiguous for real numbers, because fractional powers of negative numbers are generally not real. For an odd root of a negative number, such as the cube root of -8, use Root mode with n = 3 and result -8, which returns -2. An even root of a negative number, such as the 4th root of -16, has no real answer.
How exact are big results?
When the base and exponent are whole numbers, the result is computed exactly. Up to 60 digits it is printed in full, so 2^100 is 1267650600228229401496703205376. Longer results are shown in scientific notation with 10 significant digits, correctly rounded, such as 4.053621560e+67 for 7^80. All other results are shown to 12 significant digits.
What happens if the result is too large or too small?
Whole-number powers are exact up to about a million bits. Past that, or when a decimal result is larger than a double can hold, about 1.8e308, the calculator shows a message that the result is outside the supported range. A non-zero result too small for a double is shown as 0 with a note that it rounds to 0.
Is my data sent anywhere?
No. Every calculation runs in your browser and nothing you type is uploaded or stored. The exact big-integer arithmetic is done locally by your browser, so even very large results never touch a server.