Question

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

Answer 1

`4/5-3/5i`

Explanation:

`"multiply the numerator/denominator by the"`
`color(blue)"complex conjugate"" of the denominator"`

`"the complex conjugate of "1+2i" is "1color(red)(-)2i`

`rArr((2+1)(1-2i))/((1+2i)(1-2i))`

`"expand factors on numerator/denominator using FOIL"`

`=(2-3i-2i^2)/(1-4i^2)`

`[i^2=(sqrt(-1))^2=-1]`

`=(4-3i)/5=4/5-3/5ilarrcolor(blue)"in standard form"`

Answer 2

`4/5-(3i)/5`

Explanation:

To divide complex numbers we first remove the complex number from the denominator, by multiplying by the complex conjugate of the denominator:

This is `1-2i`

The product of a complex number and its conjugate is always a real number.

`:.`

`((1-2i)(2+i))/(( 1-2i)(1+2i))=((1-2i)(2+i))/5=(4-3i)/5=4/5-(3i)/5`