Simplify the Following
\[ \left(x+\frac{2}{x}\right)^3 + \left(x-\frac{2}{x}\right)^3 \]
Solution:
Let
\[ a=x+\frac{2}{x},\quad b=x-\frac{2}{x} \]
Using identity:
\[ a^3+b^3=(a+b)(a^2-ab+b^2) \]
\[ a+b = 2x \]
\[ ab = \left(x+\frac{2}{x}\right) \left(x-\frac{2}{x}\right) = x^2-\frac{4}{x^2} \]
\[ a^2-ab+b^2 = (a+b)^2-3ab \]
\[ = (2x)^2 -3\left(x^2-\frac{4}{x^2}\right) \]
\[ = 4x^2-3x^2+\frac{12}{x^2} \]
\[ = x^2+\frac{12}{x^2} \]
\[ a^3+b^3 = 2x\left(x^2+\frac{12}{x^2}\right) \]
\[ = 2x^3+\frac{24}{x} \]