Question

Question #189e8

Answer 1

`(3i)/(4+2i)=3/10 + 3/5i`

Explanation:

The complex conjugate or conjugate of a complex number `a+bi`, denoted `bar(a+bi)`, is given by

`bar(a+bi) = a-bi`

A useful property of the complex conjugate is that for any complex number `z`, we have `zbar(z) in RR`, that is, the product of a complex number and is conjugate is real.

We use this property by multiplying the numerator and denominator of the expression by the conjugate of the denominator. This results in a real-valued denominator when we can then distribute (if necessary).

`(3i)/(4+2i) = (3i(4-2i))/((4+2i)(4-2i))`

`=(12i-6i^2)/(16+8i-8i-4i^2)`

`=(12i-(-6))/(16-(-4))`

`=(6+12i)/20`

`=3/10 + 3/5i`