For this case we must find an expression equivalent to:[tex]64y ^ {18}-1000z ^ 6[/tex]
Then, we take common factor 8:
[tex]8 (64y ^ {18}-125z ^ 6) =[/tex]
By definition of power properties we have to:
[tex](a ^ n) ^ m = a ^ {n * m}[/tex]
So:
[tex]8 ((2y ^ 6) ^ 3- (5z ^ 2) ^ 3)[/tex]
Since both terms are perfect cubes, we factor using the cube difference formula:
[tex](a ^ 3-b ^ 3) = (a-b) (a ^ 2 + 2ab + b ^ 2)[/tex]
Where:
[tex]a = 2y ^ 6\\b = 5z ^ 2[/tex]
So:[tex]8 (2y ^ 6-5z ^ 2) ((2y ^ 6) ^ 2 + 2 (2y ^ 6) (5z ^ 2) + (5z ^ 2) ^ 2) =\\8 (2y ^ 6-5z ^ 2) (4y ^ {12} + 10y ^ 6z ^ 2 + 25z ^ 4)[/tex]
Answer:
[tex]8 (2y ^ 6-5z ^ 2) (4y ^ {12} + 10y ^ 6z ^ 2 + 25z ^ 4)[/tex]
