Cube Root of 192

The cube root of 192 is 5.76899828123. The cube root of 192 is expressed as ∛192 in radical form, where the “∛"  sign is called the “radical” sign. In exponential form, it is written as (192)^⅓. If “m” is the cube root of 192, then, m³=192.  Let us find the value of “m”.

The cube root of 192 is expressed as 4∛3 as its simplest radical form, since

 192 = 2×2×2×2×2×2×3

∛192 = ∛(2×2×2×2×2×2×3)

Group together three same factors at a time and put the remaining factor under the ∛ .

∛192= 4∛3

 We can find cube root of 192 through a method, named as, Halley’s Method. Let us see how it finds the result.
 

Now, what is Halley’s Method?

It is an iterative method for finding cube roots of a given number N, such that, x³=N, where this method approximates the value of “x”.

Formula is ∛a≅ x((x³+2a) / (2x³+a)), where 

a=given number whose cube root you are going to find

x=integer guess for the cubic root

Let us apply Halley’s method on the given number 192.

Step 1: Let a=192. Let us take x as 5, since, 125 is the nearest perfect cube which is less than 192.

Step 2: Apply the formula.  ∛192≅ 5((5³+2×192) / (2(5)³+192))=5.757 

Hence, 5.757 is the approximate cubic root of 192.
 

• Integers - Integers can be a positive natural number, negative of a positive number, or zero. We can perform all the arithmetic operations on integers. The examples of integers are, 1, 2, 5,8, -8, -12, etc.

• Whole numbers - The whole numbers are part of the number system, which includes all the positive integers from 0 to infinity. 

• Square root -The square root of a number is a value “y” such that when “y” is multiplied by itself → y × y, the result is the original number.

• Polynomial - It is an algebraic expression made up of variables like “x” and constants, combined using addition, subtraction, multiplication, or division, where the variables are raised to whole number exponents.

• Iterative method - This method is a mathematical process which uses an initial value to generate further and step-by-step sequence of solutions for a problem.