Question

Simplify and include the "if" statement

Answer 1

`\cfrac{x^(-6)-64}{4+2x^(-1)+x^(-2)}\cdot \cfrac{x^2}{4-(4)/(x)+(1)/(x^2)}~~ - ~~\cfrac{4x^2(2x+1)}{1-2x} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{x^(-6)-64}{4+2x^(-1)+x^(-2)}\implies \cfrac{(1)/(x^6)-64}{4+(2)/(x)+(1)/(x^2)}\implies \cfrac{~~ (1-64x^2 )/(x^6 ) ~~}{(4x^2+2x+1)/(x^2)}\implies \cfrac{1-64x^2}{x^4(4x^2+2x+1)} \\\\[-0.35em] ~\dotfill`

`\cfrac{x^2}{4-(4)/(x)+(1)/(x^2)}\implies \cfrac{x^2}{~~ (4x^2-4x+1 )/(x^2 ) ~~}\implies \cfrac{x^4}{x^2-4x+1} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1-64x^2}{x^4(4x^2+2x+1)}\cdot \cfrac{x^4}{x^2-4x+1}\implies \cfrac{1-64x^2}{4x^4-14x^3-3x^2-2x+1} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{now if we put that product on the left}}{\cfrac{1-64x^2}{4x^4-14x^3-3x^2-2x+1}~~ - ~~\cfrac{4x^2(2x+1)}{1-2x}}`

now, we could go further and subtract them, as you can see the LCD will be a really long one, and I'm not sure the word "simplification" would apply to that subtraction, so I'd think that's decently simplified per se, and of course, the big IF

`\textit{\LARGE IF} ~~ x\\e \begin{cases} 0\\\\ (1)/(2)\\\\ 2\pm√(3) \end{cases}`