Exponent Calculator

What this calculator does

So, you have a base and an exponent, and you want the result of raising one to the other. This tool does exactly that: enter a base and an exponent, and it returns the base raised to that power.

Both inputs accept any number, so the base or the exponent can be whole, negative, or a decimal.

How to use it

1. Enter the base, the number being raised.
2. Enter the exponent, the power to raise it to.
3. Press Calculate.

What an exponent is

An exponent is a shorthand for repeated multiplication. When the exponent is a whole number, it tells you how many times to multiply the base by itself. So 2 to the power 3 means 2 times 2 times 2, which is 8. The base is the number being multiplied, and the exponent, written as a small raised number, is the count of how many copies are multiplied together. Raising a number to a power is called exponentiation, and it is one of the basic operations of arithmetic, sitting a level up from multiplication just as multiplication sits a level up from addition.

Zero, one, and negative exponents

The idea stretches naturally past simple counting. An exponent of 1 leaves the base unchanged, since there is just one copy. An exponent of 0 gives 1 for any nonzero base, which falls out of the pattern once you notice that each step down in the exponent divides by the base. And a negative exponent means a reciprocal: 2 to the power minus 3 is 1 divided by 2 cubed, which is 1 over 8, or 0.125. The tool handles all of these, so you can explore how the result shrinks and flips as the exponent drops below zero.

Fractional exponents are roots

Here is the connection that ties this tool to all the root tools on the site. A fractional exponent is a root. Raising a number to the power one half is the same as taking its square root, and raising it to the power one third is the same as taking its cube root. So 9 to the power one half is 3, exactly the square root of 9. In general, a number to the power 1 over n is its nth root. This is why exponents and roots are really two sides of one idea, and why you can compute a root here just by entering a fractional exponent.

A worked example

Enter a base of 2 and an exponent of 3, and the tool returns 8, since 2 times 2 times 2 is 8. Enter a base of 5 and an exponent of 0, and it returns 1. Enter a base of 2 and an exponent of minus 3, and it returns 0.125. Enter a base of 9 and an exponent of 0.5, and it returns 3, the square root of 9.

Undoing an exponent

Every exponent can be undone, and there are two ways to do it depending on what you want back. If you know the result and the exponent and want the base, you take a root: that is what the root, square root, and cube root tools do. If instead you know the result and the base and want the exponent, you take a logarithm. So exponentiation, roots, and logarithms form a little family, each one reversing the raising of a base to a power from a different angle.

Questions people ask

What does an exponent do?

It tells you how many times to multiply the base by itself. Two to the power 3 is 2 times 2 times 2, which is 8.

Why does anything to the power 0 equal 1?

Because each step down in the exponent divides by the base, so stepping down to 0 from 1 divides the base by itself, giving 1. This holds for any nonzero base.

What does a negative exponent mean?

A reciprocal. Two to the power minus 3 is 1 divided by 2 cubed, which is 0.125.

What about a fractional exponent?

It is a root. A number to the power one half is its square root, and to the power 1 over n is its nth root.

How do I reverse an exponent?

Take a root to recover the base, or a logarithm to recover the exponent. The root and log tools handle these.

References

On exponents and powers. An exponent is the power to which a base is raised, and a fractional exponent is the same as a root.

1. Eric W. Weisstein, "Power," from MathWorld, a Wolfram resource, on raising a base to a power and the process of exponentiation.
2. "Nth Roots and Rational Exponents," College Algebra, Lumen Learning, on how fractional exponents correspond to roots.