complex numbers and imaginary numbers?

Question

(5 - 3(sqrt(-48)) * (2 - 4(sqrt(-27))

and


(4+i) / (3 + 5i)

Answer 1

Hi,

(5 - 3√-48)(2 - 4√-27) =

(5 - 12i√3)(2 - 12i√3) =

10 - 60i√3 - 24i√3 + 144i²(3) =

10 - 84i√3 - 432 =

-422 - 84i√3 <== answer




4.+.i
--------
3.+.5i

Multiply top and bottom both by the conjugate of the denominator, 3 - 5i.

(4.+.i)(3-5i)
----------------- =
(3.+.5i)(3-5i)

Foil these. Change i² to -1 and simplify.

12 - 20i + 3i - 5i²
----------------------- =
9 - 25i²

17 - 17i
------------
....34

Reduce by dividing everything by 17.

1 - i
------ <== answer
..2

I hope that helps!! :-)

Answer 2

Question 1
[ 5 - 3√( 48 i ² ) ] [ 2 - 4√( 27 i ² )
[ 5 - 3 (4)√3 i ] [ 2 - 4 (3) √3 i ]
[ 5 - 12 √3 i ] [ 2 - 12√3 i ]
10 - 84 √3 i - 432
- 422 - 84√3 i

Question 2
(4 + i) (3 - 5i) / 34
(12 - 17i + 5) / 34
17(1 - i) / 34
(1/2) (1 - i)

Answer 3

Complex numbers are only vectors in the complex plane. Multiplying two complex numbers is done by placing them in their polar form (magnitude and direction) and multiplying the maginitudes and adding the angles. The "real" number is like the "x" axis and the imaginary i number is like the "y" axis which is a simplification for this problem. The magnitude of the first vector is sqrt[25+9*48 ]=21.4 at -76.5 =arctan[3sr48/5]. The magnitude of the second is sr[4+16*27]=20.9 at -84.5deg Multiplying the maginitudes gives 446.8. Adding the angles gives -161deg