Carcass wrote:
\(\frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}\) =
(A) - 1
(B) 1
(C) - xy
(D) xy
(E) \(\frac{1}{xy}\)
Rather than deal with performing a variety of operations involving many fractions, a fast approach is to plug some values into the original expression.
Let's see what happens when x = 2 and y = 4
We get: \(\frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}=\frac{2^{-1} - 4^{-1} }{ (2\times4)^{-1} (2-4)}\)
Aside: \(2^{-1}=\frac{1}{2}=0.5\), \(4^{-1}=\frac{1}{4}=0.25\) and \(8^{-1}=\frac{1}{8}=0.125\)
So we get: \(=\frac{0.5 - 0.25 }{ (8)^{-1} (-2)}\)
\(=\frac{0.5 - 0.25 }{ (0.125)(-2)}\)
\(=\frac{0.25 }{ -0.25}=-1\)
Since answer choice A is the only answer choice that evaluates to be -1 when x = 2 and y = 4, the correct answer is A
Cheers,
Brent