Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (5x26 • x) -  8  = 0 
 Step  2  :

Trying to factor as a Difference of Cubes:

 2.1      Factoring:  5x²⁷-8 

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 :  5  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  2  :

  5x27 - 8  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    5x²⁷-8 = 0 

 Add  8  to both sides of the equation : 
                      5x²⁷ = 8
Divide both sides of the equation by 5:
                     x²⁷ = 8/5 = 1.600
                     x  =  27th root of (8/5) 

 The equation has one real solution
This solution is  x = 27th root of ( 1.600) = 1.01756

One solution was found :