Root Calculator

What this calculator does

So, you want the nth root of a number: the square root, the cube root, the fourth root, or any higher one. This tool finds it. You give it the number and the degree of the root, and it returns the value, rounded to four decimal places.

There are two inputs: the number a, and the degree n, which is the "how many" of the root. The degree needs to be at least 2.

How to use it

1. Enter the number (a).
2. Enter the degree (n), 2 or more.
3. Press Calculate.

What an nth root is

The nth root of a number answers a simple question: what number, raised to the power n, gives you back the original? The cube root of 27 is 3, because 3 to the power 3 is 27. The fourth root of 16 is 2, because 2 to the power 4 is 16. The degree n is the power you are undoing. A degree of 2 gives the square root, a degree of 3 gives the cube root, and any larger degree gives the corresponding higher root. The tool computes this as the number raised to the power 1 over n, since a fractional exponent and a root are the same thing.

The reverse of raising to a power

A root is the exact opposite of an exponent. Raising to a power takes a number and multiplies it by itself a set number of times; taking a root starts from the result and works backwards to the number you began with. So if 2 to the power 5 is 32, then the fifth root of 32 is 2, right back to the start. This makes roots the natural partner of the exponent calculator: one builds a power up, the other takes it apart.

When there is no real answer

Not every root of every number exists among the ordinary real numbers, and the tool is honest about this. If you ask for an even root, such as a square or fourth root, of a negative number, there is no real value that works, because multiplying a real number by itself an even number of times can never give a negative. In those cases the tool reports that there is no real number satisfying the criteria, rather than inventing an answer. An odd root of a negative number is different: it does have a real value. The cube root of -8 is -2, because -2 cubed is -8, so the tool returns that. Only even roots of negatives fall outside the real numbers.

A worked example

Enter a number of 27 and a degree of 3, and the tool returns 3, since 3 cubed is 27. Enter a number of 16 and a degree of 4, and it returns 2, since 2 to the fourth is 16. Enter a number of 10 and a degree of 2, and it returns about 3.1623, the square root of 10, which is not a whole number.

Roots you will meet most

Two degrees come up so often they have their own dedicated tools. The square root, degree 2, is the most common of all, and the square root calculator is tuned just for it. The cube root, degree 3, is the next most familiar, with its own cube root calculator. This general tool covers those and every higher degree besides. If you would rather see the answer written in proper radical notation, with the root sign and its index, the radical calculator presents the same value that way.

Questions people ask

What does the nth root give me?

The number that, raised to the power n, produces your original number. The cube root of 27 is 3 because 3 cubed is 27.

What is the degree?

The power you are undoing. Degree 2 is a square root, degree 3 a cube root, and higher degrees the matching higher roots. It must be at least 2.

How does a root relate to an exponent?

It is the reverse. A root undoes raising to a power, so the nth root is the number raised to the power 1 over n.

Why do I sometimes get no answer?

An even root of a negative number has no real value, because an even number of multiplications can never yield a negative. The tool says so rather than guessing.

Where are the square and cube root tools?

The square root and cube root have their own dedicated calculators. This one handles those degrees and every higher one.

References

On the nth root. The nth root is the inverse of raising to a power, equal to the number raised to the power 1 over n.

1. Eric W. Weisstein, "nth Root," from MathWorld, a Wolfram resource, on the nth root as the inverse function of taking a power.
2. Christopher Stover and Eric W. Weisstein, "Radical," from MathWorld, a Wolfram resource, on the equivalence of roots and fractional exponents.