Cube root

A cube root is a number that can be multiplied three times to give us the value under the radical symbol (also referred to as the radix). A cube root (or any root) is denoted as follows:

The only difference between the denotation of a cube root and any other root, referred to as an nth root, is the index. The index changes to indicate the number of times the root is multiplied to equal the radicand. We can also think of the cube root as:

 z3 = x
eg:  23 = 8

We can read the above example as "the cube root of 8 is 2."

Unlike square roots, it is possible to take the cube root of negative numbers. In general, it is possible to take the nth root of a negative number as long as n is odd. If n is even, we can't take the root without using imaginary numbers since a negative number raised to an even exponent is positive. The cube root of a negative number is the same as the cube root of its positive value, just negative. For example, . On the other hand, -3 cubed is:

-3 × -3 × -3 = -27, so

Finding cube roots

Cube roots are relatively simple if the radicand is a perfect cube. Like square roots, or any other radical, we usually try to simplify cube roots by manipulating the expression such that we are left with expressions involving a product of perfect cube(s) where possible. If this is not possible, we can only estimate the value of the cube root, though like all nth roots, it is very difficult to estimate the roots of non-perfect cubes, and this is usually done using a calculator.


Find :

It is helpful to remember some of the perfect cubes in order to be able to work with cube roots. Below is a table of the cubes of 0-20.

 #  Perfect cube
0 0
1 1
2 8
3 27
4 64
5 125
6 216
7 343
8 512
9 729
10 1000
11 1331
12 1728
13 2197
14 2744
15 3375
16 4096
17 4913
18 5832
19 6859
20 8000