Question

The expression 3+i / 4+i is equivalent to

Answer 1

Multiply 3+i / 4+i by 1 by using the conjugate of 4+i over itself (4-i / 4-i)
(3+i)/(4+i) × (4-i)/(4-i)
(12-3i+4i-
`i^(2)`) / (16-4i+4i-
`i^(2)`)
(12+i-
`i^(2)`) / (16 -
`i^(2)`)
(12+i-(-1)) / (16-(-1))
(12+i+1) / (16+1)
13+i / 17
3+i / 4+i =
`(13)/(17) + (1)/(17)i`

Answer 2

`i=√(-1)\to i^2=-1\\========================\\(3+i)/(4+i)=(3+i)/(4+i)\cdot(4-i)/(4-i)=((3+i)(4-i))/((4+i)(4-i))=((3)(4)-(3)(i)+(i)(4)-(i)(i))/(4^2-i^2)\\\\=(12-3i+4i-(-1))/(16-(-1))=(13+i)/(17)=\boxed{(13)/(17)+(1)/(17)i}`