Question

If $x=-2$ and $y=1,$ by using an identity find the value of the following:

$\left(ii\right)$ $(\frac{2}{x}-\frac{x}{2})(\frac{4}{x^2}+\frac{x^2}{4}+1)$

Answer 1

Given: $(\frac{2}{x}-\frac{x}{2})(\frac{4}{x^2}+\frac{x^2}{4}+1)$

Using Identity.

$a^3-b^3=(a-b)(a^2+ab+b^2)$

Comparing given equation with identity we get.

$\therefore {\left(\frac{2}{x}\right)}^3-{\left(\frac{x}{2}\right)}^3$

$=(\frac{2}{x}-\frac{x}{2})({\left(\frac{2}{x}\right)}^2+{\left(\frac{x}{2}\right)}^2+\frac{2}{x}\times \frac{x}{2})$

$\therefore (\frac{2}{x}-\frac{x}{2})(\frac{4}{x^2}+\frac{x^2}{4}+1)=\frac{8}{x^3}-\frac{x^3}{8}$

Substituting $x=-2$ we get,

$=\frac{8}{(-2{)}^3}-\frac{(-2{)}^3}{8}$

$=\frac{8}{-8}-\frac{-8}{8}$

$=-1-(-1)$

$=-1+1$

$=0$