Find the multiplicative inverse of the complex number 4 - 2i .

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You need to remember that you only may evaluate the multiplicative inverse of a complex number in its rectangular form, such that:

Multiplicative inverse of `a - b*i` is `1/(a -b*i)`

Reasoning by analogy yields:

Multiplicative inverse of `4 - 2*i` is `z' = 1/(4 - 2i).`

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You need to remember that you only may evaluate the multiplicative inverse of a complex number in its rectangular form, such that:

Multiplicative inverse of `a - b*i` is `1/(a -b*i)`

Reasoning by analogy yields:

Multiplicative inverse of `4 - 2*i` is `z' = 1/(4 - 2i).`

You need to perform the following multiplication, such that:

`1/(4 - 2i) = (4+2i)/((4 + 2i)(4 - 2i))`

Converting the product `((4 + 2i)(4 - 2i))` into a difference of squares, yields:

`1/(4 - 2i) = (4+2i)/(4^2 - (2i)^2)`

Using complex number theory you need to substitute `-1` for `i^2` , such that:

`1/(4 - 2i) = (4+2i)/(16 + 4)`

`1/(4 - 2i) = (4+2i)/20 => 1/(4 - 2i) = (2(2+i))/20`

Reducing duplicate factors yields:

`1/(4 - 2i) = (2+i)/10`

Hence, evaluating the multiplicative inverse of the given complex number `4 - 2i` yields `(2+i)/10.`

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