Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "y1"   was replaced by   "y^1". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (8 • (y2)) -  (2•33y117)  = 0 
 Step  2  :

Equation at the end of step  2  :

  23y2 -  (2•33y117)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   8y² - 54y¹¹⁷  =   -2y² • (27y¹¹⁵ - 4) 

Equation at the end of step  4  :

  -2y2 • (27y115 - 4)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 5.2      Solve  :    -2y² = 0 

 Multiply both sides of the equation by (-1) :  2y² = 0


Divide both sides of the equation by 2:
                     y² = 0
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      y  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  y = 0

Solving a Single Variable Equation :

 5.3      Solve  :    27y¹¹⁵-4 = 0 

 Add  4  to both sides of the equation : 
                      27y¹¹⁵ = 4
Divide both sides of the equation by 27:
                     y¹¹⁵ = 4/27 = 0.148
                     y  =  115th root of (4/27) 

 The equation has one real solution
This solution is  y = 115th root of ( 0.148) = 0.98353

Two solutions were found :

1.  y = 115th root of ( 0.148) = 0.98353
2.  y = 0