Cube root

Cubic root . It is the quantity that has to be multiplied by itself 3 times to obtain as a result the radicand or number that accompanies the mathematical root symbol

Cube of a number


In arithmetic and algebra, the cube of a number x equals the third power, which is corresponding to the result of multiplying a number by itself three times.

Real numbers

Usually a real number has three cube roots, one root corresponds to a real and the other two to complex numbers. A complex number represents the addition of a real number to an imaginary number .

Example with the cube roots of the number 8

If x and y are real numbers, then there will be a single solution such that the equation also has a single solution, this will correspond to a real. Using this definition, the cube root of a negative number is also a negative number. Thus, the principle of the cube root of x is constituted by: If x and y are complex, it can be said that there are three solutions (as long as x is not null) and thus x has three cube roots, one is real root and the others are complex, in the form of a conjugate pair.

Example of the roots of number one

These two roots are related to all the other cube roots of other different numbers. So if a number is corresponding to the cube root of a real number, the cube roots can be calculated by multiplying the number by the roots of the cube root of one.

If complex numbers are referenced, the main value of the cube roots is defined like this:


Where ln (x) is the natural logarithm. If x is written as:


r is corresponding to a positive real number and θ falls in the range:

So the cube root is as follows:



This means that if we take the cube root of a complex, we are taking the cube root of the radius and the polar angle is then dividing into three parts, so that the three roots are defined. With this definition, the cube root of a negative number will be a complex number.

Calculate the cube root of a number

1. The cube root is calculated in a similar way to the square root , but the digits of the radicand are separated into groups of three figures. For example: 16387064 would be separated into: 16’387’064. 2. Then we proceed to calculate an integer that, when it is raised to the cube, comes as close as possible to the number of the first group (always starting from the left).

In the example, the first number is 16 and the integer that cubed closer to 16 is number 2. So 2 is the first number in the root.

3. This number is then cubed and the number in the first group is immediately subtracted.

In the example 23 = 8 and subtracting it from the number of the first group that is 16 we obtain that 16 – 8 = 8.

4. The number of the next group is placed next to the rest previously obtained.

In the example, the following figure remains: 8387.

5. Subsequently, it is necessary to calculate a number a, executing the operations:

Approaching as closely as possible to the number obtained in point 4. The number a will be the next digit of the root. In the example this corresponds to number 5.

6. Now this number is subtracted from the number we obtained in point 4. Therefore: 8387- 7625 = 762. 7. Point 4 is done again. In the example: 762064 8. The fifth step and the number are repeated obtained will be the subsequent number of the root.

In the example it would be 4, because


9. We proceed to make the sixth point again. In the example 762064-762064 = 0

The root is exact since the rest is zero.

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