Question

If $x=3$ and $y=-1,$ find the values of the following using identity:

$\left(v\right)$ $(\frac{5}{x}+5x)(\frac{25}{x^2}-25+25x^2)$

Answer 1

Given, $(\frac{5}{x}+5x)(\frac{25}{x^2}-25+25x^2)$

Using Identity,

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

Comparing given equation with identity we get.

$\therefore {\left(\frac{5}{x}\right)}^3+(5x{)}^3$

$=(\frac{5}{x}+5x)({\left(\frac{5}{x}\right)}^2-\frac{5}{x}\times 5x+(5x{)}^2)$

$\therefore (\frac{5}{x}+5x)(\frac{25}{x^2}-25+25x^2)=\frac{125}{x^3}+125x^3$

Substituting $x=3,y=-1$ we gwt,

$=\frac{125}{3^3}+125\times 3^3$

$=\frac{125}{27}+125\times 27$

$=\frac{125}{27}+3375$

$=\frac{125+91125}{27}$

$=\frac{91250}{27}$

$\therefore (\frac{5}{x}+5x)(\frac{25}{x^2}-25+25x^2)=\frac{91250}{27}$