Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x9"   was replaced by   "x^9". 

Rearrange:

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

                     x^215*x-(x^98)=0 

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   x²¹⁶ - x⁹⁸  =   x⁹⁸ • (x¹¹⁸ - 1) 

Trying to factor as a Difference of Squares :

 2.2      Factoring:  x¹¹⁸ - 1 

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 : 1 is the square of 1
Check :  x¹¹⁸  is the square of  x⁵⁹ 

Factorization is :       (x⁵⁹ + 1)  •  (x⁵⁹ - 1) 

Equation at the end of step  2  :

  x98 • (x59 + 1) • (x59 - 1)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    x⁹⁸ = 0 

  Solution is  x⁹⁸ = 0

Solving a Single Variable Equation :

 3.3      Solve  :    x⁵⁹+1 = 0 

 Subtract  1  from both sides of the equation : 
                      x⁵⁹ = -1
                     x  =  59th root of (-1) 

 Negative numbers have real 59th roots.
 59th root of (-1) = ⁵⁹√ -1• 1  = ⁵⁹√ -1 • ⁵⁹√ 1  =(-1)•⁵⁹√ 1 

The equation has one real solution, a negative number This solution is  x = negative

Solving a Single Variable Equation :

 3.4      Solve  :    x⁵⁹-1 = 0 

 Add  1  to both sides of the equation : 
                      x⁵⁹ = 1
                     x  =  59th root of (1) 

 The equation has one real solution
This solution is  x =

Three solutions were found :

1.  x =
2.  x = negative
3.  x⁹⁸ = 0