Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22w212 • w) -  44  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   4w²¹³ - 44  =   4 • (w²¹³ - 11) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  w²¹³ - 11 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  11  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  3  :

  4 • (w213 - 11)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    4   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    w²¹³-11 = 0 

 Add  11  to both sides of the equation : 
                      w²¹³ = 11
                     w  =  213th root of (11) 

 The equation has one real solution
This solution is  w = 213th root of 11 = 1.0113

One solution was found :