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# Cube Root

## Definition of Cube Root

To indicate a cube root, we use the same radical symbol we used for square roots, but we write a small 3 just above the on the radical symbol to indicate a cube root. The small 3 is called the index of the radical.

For example, the cube root of 64 is written  Definition â€” Cube Root

The cube root of a real number, a, is written If b is a real number and b3 = a, then Example: because 43 = 64.

Here are some examples: because 53 = 5 Â· 5 Â· 5 = 125 because 13 = 1 Â· 1 Â· 1 = 1. = 1 because (-1)3 = (-1)(-1)(-1) = -1. because (-2)3 = (-2)(-2)(-2) = -8.

Note:

To find the cube root of a number, we reverse the operation of cubing.

For example, to find we ask â€œWhat number cubed is 8?â€.

Therefore, The last two examples show that a cube root, unlike a square root, can have a negative radicand.

We can use geometry to provide a visual interpretation of a cube root.

For example, suppose a cube has a volume of 125 cubic inches. The length of each side is the cube root of the volume.

That is, the length of a side of the cube = = 5 in.

A perfect cube is a number that has a rational cube root.

For example, 125 is a perfect cube because 53 = 125.

As we work with cube roots, we will find it useful to recognize perfect cubes and their cube roots.

Now we will summarize the relationship between cubes and cube roots.

Property â€” Cubes and Cube Roots

English Cubing and taking a cube root â€œundoâ€ each other.

Algebra If a is a real number, then Example Note:

Unlike square roots, there are no â€œprincipalâ€ cube roots.

The sign of a cube root depends on the sign of its radicand. Perfect Squares Principal Square Roots 03 = 0 13 = 1 23 = 4 33 = 9 43 = 16 53 = 25 63 = 36 73 = 49 83 = 64 93 = 81 103 = 100   Property â€” Cubes and Cube Roots

English Cubing and taking a cube root â€œundoâ€ each other.

Algebra If a is a real number, then Example 