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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 — 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 · 5 · 5 = 125

because 13 = 1 · 1 · 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 “What number cubed is 8?”.

The answer is 2.

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 — Cubes and Cube Roots

English Cubing and taking a cube root “undo” each other.

Algebra If a is a real number, then

Example

Note:

Unlike square roots, there are no “principal” 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 — Cubes and Cube Roots

English Cubing and taking a cube root “undo” each other.

Algebra If a is a real number, then

Example

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