Multiplicative Inverse

If \alpha=2+i, then in the form of a+bi, what does \alpha^{-1} equal?
Source: NCTM Mathematics Teacher, September 2006

Solution
By definition of multiplication inverse
\alpha\alpha^{-1}=1
\alpha^{-1}=\dfrac{1}{2+i}
=\dfrac{1}{2+i}\times\dfrac{2-i}{2-i}
=\dfrac{2-i}{2^2-i^2}
=\dfrac{2-i}{4+1}
=\dfrac{2-i}{5}
=\dfrac{2}{5}-\dfrac{1}{5}i

Answer: \dfrac{2}{5}-\dfrac{1}{5}i

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About mvtrinh

Retired high school math teacher.
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