Question

How do you simplify `1/(2+3i)`?

Answer 1

`1/(2+3i)=2/13-3/13i`

Explanation:

`1/(2+3i)` is reciprocal of complex number `2+3i`. To find reciprocal we multiply numerator and denominator each by complex conjugate of `2+3i`, which is `2-3i`. Hence,

`1/(2+3i)=1/(2+3i)×(2-3i)/(2-3i)`

= `(2-3i)/(2^2-(3i)^2)`

= `(2-3i)/(4-9i^2)`

= `(2-3i)/(4+9)`

= `2/13-3/13i`

Answer 2

`2/13-3/13i`

Explanation:

`"multiply the numerator/denominator by the complex"`
`"conjugate of the denominator"`

`"the conjugate of "2+3i" is "2color(red)(-)3i`

`(2+3i)(2-3i)=2^2-9i^2=4+9=13`

`=(2-3i)/13=2/13-3/13ilarrcolor(blue)"in standard form"`