Simplify the Following
\[ (2x-5y)^3-(2x+5y)^3 \]
Solution:
Let
\[ a=2x-5y,\quad b=2x+5y \]
Using identity:
\[ a^3-b^3=(a-b)(a^2+ab+b^2) \]
\[ a-b = (2x-5y)-(2x+5y) = -10y \]
\[ ab = (2x-5y)(2x+5y) = 4x^2-25y^2 \]
\[ a+b = 4x \]
\[ a^2+ab+b^2 = (a+b)^2-3ab \]
\[ = (4x)^2-3(4x^2-25y^2) \]
\[ = 16x^2-12x^2+75y^2 \]
\[ = 4x^2+75y^2 \]
\[ a^3-b^3 = (-10y)(4x^2+75y^2) \]
\[ = -40x^2y-750y^3 \]