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Last updated on November 30th, 2024

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Cube Root of 192

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Foundation
Intermediate
Advance Topics

The cube root of 192 is the value that, when multiplied by itself three times (cubed), gives the original number 192. Do you know? Cube roots apply to our real life also, like that for measuring volume and scaling, density and mass calculation, etc.

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What Is the Cubic Root of 192?

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

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Finding the Cubic Root of 192

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


 192 = 2×2×2×2×2×2×3


βˆ›192 = βˆ›(2×2×2×2×2×2×3)


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


βˆ›192= 4βˆ›3


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

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Cubic Root of 192 By Halley’s Method

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 βˆ›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.  βˆ›192≅ 5((53+2×192) / (2(5)3+192))=5.757 


Hence, 5.757 is the approximate cubic root of 192.
 

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Important Glossaries for 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.
     
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