If i^2=-1, then (i-i^-1)^-1=

1994 AMC 12 Problem 12

Attempt Problem 12 of the 1994 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1994 AMC 12 solutions, or check the answer key.

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12.

i^2=-1,

then

\left(i-i^{-1}\right)^{-1}=

0

-2i

2i

-\frac{i}{2}

\frac{i}{2}

Use

i^{-1}=-i

After simplifying the parentheses, rationalize

\frac{1}{2i}

Because

i^{-1}=-i,

\begin{aligned} \left(i-i^{-1}\right)^{-1} &=(i+i)^{-1}\\ &=\frac1{2i}\\ &=-\frac i2. \end{aligned}

Thus the correct answer is D.