Cube root is one of three equal factors of a number (see Factor). The same number (m) taken as a factor three times is the cube root of another number (n). Thus, m X m X m = n. For example, 2 is the cube root of 8, because 2 X 2 X 2 = 8, and -5 is the cube root of -125, because -5 X -5 X -5 = -125. A real number has only one real cube root, which is positive or negative, according to whether the given number is positive or negative. When a cube root or any other root of a number is to be extracted (determined), another symbol is placed over the number. This symbol is called the root sign, or radical sign. If the root to be extracted is a cube root, a small figure 3 is added to the root sign.
To find the cube root of a number, you can use a scientific calculator, or you can look up the root in a table of cube roots. If neither of these is available, you must calculate the root.
You can use a procedure called Newton’s method to calculate the cube root of a number between 1 and 1,000. For example, you might wish to find the cube root of 200. Since 5 X 5 X 5 = 125, and 6 X 6 X 6 = 216, it is easy to see that 6 is the closest integral, or whole number, cube root of 200. A closer complete approximation can be made by dividing 200 by the square of 6, or 6 X 6, which equals 36. To the nearest tenth, this gives 5.6. Thus, 6 X 6 X 5.6 is approximately 200.
To get the second approximation of the cube root of 200, average the three factors 6, 6, and 5.6. This will give (6 + 6 + 5.6)/3 = 5.9. This procedure is repeated to obtain a still better approximation. Thus, 200/(5.9 X 5.9) = 200/34.81 = 5.74, and the next approximation is given by (5.9 + 5.9 + 5.74)/3 = 5.85. Repeating once more gives 200/(5.85 X 5.85) = 200/34.2225 = 5.8441, which gives the next approximation (5.85 + 5.85 + 5.8441)/3 = 5.8480.
This process may be continued indefinitely. In each approximation beyond the second, you can retain a number of digits that is one less than twice the number of digits found in the previous approximation. For example, the second approximation, 5.9, contains two digits. The third approximation may retain three digits, and the fourth approximation may retain five digits.
If the number whose cube is desired is not between 1 and 1,000, either multiply or divide it successively by 1,000 to bring it within this range. The cube root of this number will lie between 1 and 10. After finding the cube root, either divide or multiply it successively by 10 as many times as necessary to give the cube root of the original number.
