Expert: Paul Klarreich - 4/1/2008

Question
Howdy Paul, here are four questions I have & if you can show me the steps to the answers I will send you a winning lottery ticket.  Well I can't be 100% sure its a winner, but I have a good feeling about it.  OK here they is:
Write the complex number in standard form:
1.) 3i(2+5i) + 2i(3-4i)
2.) (6 + 5i)(2 - 5i)

Write expression as a complex number in standard form:
1.) (4 + i)/(3 + 5i)

Lastly; write the following complex number in trigonometric
form:
1.) z=1 + i(square root of 3)

I thank you 'forehand kind sir  

Answer
Questioner:   Rich
Category:  Advanced Math
Private:  No
 
Subject:  Complex Numbers
Question:  Howdy Paul, here are four questions I have & if you can show me the steps to the answers I will send you a winning lottery ticket.  Well I can't be 100% sure its a winner, but I have a good feeling about it.  OK here they is:
Write the complex number in standard form:
1.) 3i(2+5i) + 2i(3-4i)
2.) (6 + 5i)(2 - 5i)

Write expression as a complex number in standard form:
1.) (4 + i)/(3 + 5i)

Lastly; write the following complex number in trigonometric
form:
1.) z = 1 + i(square root of 3)

I thank you 'forehand kind sir
..................................
Hi, Rich,

Write the complex number in standard form:

All you have to do here is multiply out the algebra, simplify, and when you see i^2, replace it with (-1)  [Yes, with the parentheses.  If you leave them out, you lose points, even if your final answer is correct.] and keep simplifying.

1.) 3i(2+5i) + 2i(3-4i)
6i + 15i^2 + 6i - 8i^2
6i + 15(-1) + 6i - 8(-1)
12i - 15 + 8 = -7 + 12i


2.) (6 + 5i)(2 - 5i)

12 + 10i - 30i - 25i^2

12 + 10i - 30i - 25(-1)

etc.
......................

Write expression as a complex number in standard form:
1.) Rationalize the denominator:
(4 + i)   (3 - 5i)
--------- --------- =
(3 + 5i)  (3 - 5i)

12 + 3i - 20i - 5i^2
--------------------
 9 - 25i^2

12 + 3i - 20i - 5(-1)
--------------------
 9 - 25(-1)

17 - 17i
---------
 34

1/2 - (1/2)i

.........................
1.) z = 1 + i(square root of 3)

x = 1, y = sqrt(3)

Find your modulus (radius) :

r^2 = 1^2 + (sqrt(3))^2
r = 2

Use the trig you learned in high school (go back there if you have to) to get

theta = 60 degrees or  pi/3.

Now  z = 2 cis(pi/3), or whatever notation your teacher likes.