+9673 1)Simplify \(\dfrac{x^{-1}+y^{-1}}{x+y}\)
2)Simplify \(\dfrac{x^{-1}-y^{-1}}{x-y}\)
3)Simplify \(\dfrac{x^{-2}+y^{-2}}{x+y}\)
4)Simplify \(\dfrac{x^{-2}-y^{-2}}{x-y}\)
These questions are 'created' by my teacher, I find this kind of funny and want to share this to you!! :D
+9673 I would like you to share your answers to these questions.
I will go first :)
1) \(\dfrac{x^{-1}+y^{-1}}{x+y}\\ =\dfrac{(x^{-1}+y^{-1})(xy)}{(x+y)(xy)}\\ =\dfrac{y + x}{(x+y)(xy)}\\ = \dfrac{1}{xy}\)
2)\(\dfrac{x^{-1}-y^{-1}}{x-y}\\ =\dfrac{(x^{-1}-y^{-1})(xy)}{(x-y)(xy)}\\ = \dfrac{-(x-y)}{(x-y)(xy)}\\ =-\dfrac{1}{xy}\)
3)\(\dfrac{x^{-2}+y^{-2}}{x+y}\\ =\dfrac{\frac{1}{x^2}+\frac{1}{y^2}}{x+y}\\ = \dfrac{\frac{x^2+y^2}{(xy)^2}}{x+y}\\ =\dfrac{x^2+y^2}{(xy)^2(x+y)}\)
4)\(\dfrac{x^{-2}-y^{-2}}{x-y}\\ =\dfrac{(x^{-1}+y^{-1})(x^{-1}-y^{-1})}{x-y}\\ =\dfrac{x^{-1}-y^{-1}}{x-y}(x^{-1}+y^{-1})\\ =-\dfrac{x^{-1}+y^{-1}}{xy}\)
The last step of 4) is from 2)!!
