Question

How do you simplify `(2-3i)(4+i)`?

Answer 1

To simplify this expression, multiply top and bottom by the complex conjugate, `(4-i)`. This yields `(11-4i)/17`

Explanation:

To simplify an expression like this, we multiply top and bottom of the fraction (denominator and numerator) by the complex conjugate of the bottom (numerator). For a complex expression `(a+bi)`, the complex conjugate is `(a-bi)`. When we do the calculation, it will become clear why this works so well.

Note that `(a-bi)/(a-bi)` is just the same as 1, so when we do this multiplication the result is the same number we started with.

`(2-3i)/(4+i)*(4-i)/(4-i) = (8-12i-2i+3i^2)/(16+4i-4i-i^2)`

But since `i=sqrt-1`, `i^2=-1`. Using this and collecting like terms:

`(8-14i-3)/(16+1)=(11-4i)/17`