Question

How do you write the following quotient in standard form `1/(4-5i)^2`?

Answer 1

`-9/1681+40/1681i`

Explanation:

`"expand the denominator using FOIL method"`

`•color(white)(x)i^2=(sqrt(-1))^2=-1`

`rArr(4-5i)^2=16-40i+25i^2=-9-40i`

`rArr1/(4-5i)^2=1/(-9-40i)`

`"to rationalise the denominator multiply "`
`"numerator/denominator"`

`"by the "color(blue)"conjugate " "of "-9-40i`

`"the conjugate of "-9-40i" is "-9color(red)(+)40i`

`rArr(-9+40i)/((-9-40i)(-9+40i))`

`=(-9+40i)/1681`

`=-9/1681+40/1681ilarrcolor(blue)"in standard form"`