Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (52k215 • k) -  41
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  25k²¹⁶-41 

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 :  25  is the square of  5 
Check : 41 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:  25k²¹⁶-41 

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 :  25  is not a cube !!

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

Final result :

  25k216 - 41