Question

Please simplify these and explain thoroughly what you did. The problems are in the picture attached. I believe them to be complex numbers. Thank you for your help and time.

Answer 1

1\ 8.66i

2\ -i

Explanation:

1\

`√( - 75 ) = √(75) * √( - 1) \\ and √( - 1) = i \\ = 8.66i`

2\

`{i}^(7) = {i}^(6) * i \\ = {i}^(2 ) * i ^(4) * i \\ = - 1 * { - 1}^(2) * i \\ = - 1 * 1 * i \\ = - i`

3\

Using the same steps..

Answer 2

`5i√(3)`

- i

i

Explanation:

1

`√(-75)`

take out i which is (
`√(-1)`):

`√(75) *√(-1) = i√(75)`

`√(75) = √(5*5*3) = 5√(3)`

`i√(75) = 5i√(3)`

2

`i^(7) = i^(4) * i^(3) = 1* -i = -i\\\\i^(3) = i^(2) * i = -1 * i = -i`

(this above, was just a quick explanation on how i^3 is -i)

3

`i^(49) = i`

lets use the reference of
`i^(4) = 1` :

49/4= 12 remainder 1

Using above knowledge

`i^(49)` =

`(i^(4))^(12) * i^(1)`

(1)12 × i =

1 × i =

i