Cube And Cube Root Calculator

What this calculator does

So, you want both the cube of a number and its cube root at the same time. This tool gives you the pair. Enter one number and it returns two results: the number cubed, and the cube root of the number, each to ten decimal places.

There is one input. Seeing the cube and the cube root side by side is the whole point, because they are opposite operations.

How to use it

1. Enter your number, which should be zero or positive.
2. Press Calculate to see its cube and its cube root together.

Two operations, one number

The tool does two things to your number. To cube it, it multiplies the number by itself three times, so 4 cubed is 4 times 4 times 4, which is 64. To take its cube root, it finds the number that would have to be cubed to produce your number, so the cube root of 4 is about 1.587, since that value cubed comes back to 4. One operation builds up, raising to the third power; the other works backwards, undoing a third power. Both start from the same input.

Cube and cube root undo each other

The reason it is useful to show them together is that they are exact inverses, and seeing the pair makes that plain. Cube a number and then take the cube root of the result, and you land back on the number you started with. Take the cube root first and then cube it, and again you return to the start. The two operations cancel out, which is exactly the relationship between any power and its matching root. This tool puts both halves of that relationship in front of you at once.

Perfect cubes and the pattern

Numbers like 1, 8, 27, 64, and 125 are called perfect cubes, because they are whole numbers cubed: 1 is 1 cubed, 8 is 2 cubed, 27 is 3 cubed, and so on. For these, the cube root comes out as a tidy whole number. For everything in between, the cube root is a decimal. Feeding a series of numbers through the tool is a quick way to see how the cubes shoot up steeply while the cube roots climb gently, the two columns pulling in opposite directions from the same starting values.

A worked example

Enter 8. The tool returns a cube of 512, since 8 cubed is 512, and a cube root of 2, since 2 cubed is 8. Enter 3 and it returns a cube of 27 and a cube root of about 1.4422495703. In each case the two answers show the same number pushed in opposite directions.

Questions people ask

What does this tool give me?

Two results from one number: the number cubed, and the cube root of the number.

What does cubing a number mean?

Multiplying it by itself three times. Four cubed is 4 times 4 times 4, which is 64.

How are the cube and cube root related?

They are opposite operations. Cubing a number and then taking the cube root returns you to the original, and the reverse order does too.

What is a perfect cube?

A whole number that is another whole number cubed, such as 8, 27, or 64. Its cube root is a whole number.

What if I only want one of them?

The cube root calculator gives just the cube root, and the exponent calculator will cube a number using an exponent of 3.

References

On the cube and cube root. Cubing raises a number to the third power, and the cube root reverses it, the two being inverse operations.

1. Eric W. Weisstein, "Cube Root," from MathWorld, a Wolfram resource, on the cube root as the number whose cube gives the original.
2. Eric W. Weisstein, "Power," from MathWorld, a Wolfram resource, on raising a number to a power, of which cubing is the third.