Solution - Nonlinear equations

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x^210*13-(4)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  13x210 -  4  = 0 

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  13x²¹⁰-4 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  13  is not a square !!

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

Trying to factor as a Difference of Cubes:

 2.2      Factoring:  13x²¹⁰-4 

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 :  13  is not a cube !!

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

Equation at the end of step  2  :

  13x210 - 4  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    13x²¹⁰-4 = 0 

 Add  4  to both sides of the equation : 
                      13x²¹⁰ = 4
Divide both sides of the equation by 13:
                     x²¹⁰ = 4/13 = 0.308
                     x  =  210th root of (4/13) 

 The equation has two real solutions  
 These solutions are  x = 210th root of ( 0.308) = ± 0.99440  
 

Two solutions were found :