1/ab *{( a²+b²)/(a-b)}^2-a/b-b/a-2
You asked:
Evaluate the expression: \(\frac{1}{a b} {\left( \frac{{a}^{2} + {b}^{2}}{a - b} \right)}^{2} - \frac{a}{b} - \frac{b}{a} - 2\)
MathBot Answer:
Evaluated
\(\displaystyle \frac{1}{a b} {\left( \frac{{a}^{2} + {b}^{2}}{a - b} \right)}^{2} - \frac{a}{b} - \frac{b}{a} - 2 = - \frac{a}{b} - 2 - \frac{b}{a} + \frac{\left(a^{2} + b^{2}\right)^{2}}{a b \left(a - b\right)^{2}} \)
Expanded
\[\frac{1}{a b} {\left( \frac{{a}^{2} + {b}^{2}}{a - b} \right)}^{2} - \frac{a}{b} - \frac{b}{a} - 2 = \frac{a^{3}}{a^{2} b - 2 b^{2} a + b^{3}} + \frac{2 a b}{a^{2} - 2 a b + b^{2}} - \frac{a}{b} + \frac{b^{3}}{a^{3} - 2 a^{2} b + b^{2} a} - 2 - \frac{b}{a}\]
Factored
\[\frac{1}{a b} {\left( \frac{{a}^{2} + {b}^{2}}{a - b} \right)}^{2} - \frac{a}{b} - \frac{b}{a} - 2 = \frac{4 a b}{\left(a - b\right)^{2}}\]