5-9i/5+2i
First, multiply both the numerator (top) and denominator (bottom) by the equivalent of 1 to eliminate i in the denominator:
(5 - 9i)(5 - 2i)
----------------
(5 + 2i)(5 - 2i)
25 - 55i + 18i^2
----------------
25 - 4i^2
Remember that i^2 = -1
25 - 55i + 18(-1)
-----------------
25 - 4(-1)
25 - 55i - 18
-------------
25 + 4
7 - 55i
-------
29
You can change this to standard form if asked to do so.
I hope this helps.
To divide complex numbers like 5-9i divided by 5+2i, we can use a method called "rationalizing the denominator." Here are the steps:
Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 5+2i is 5-2i. So, we multiply both the numerator and denominator by 5-2i.
(5-9i) * (5-2i) / (5+2i) * (5-2i)
Step 2: Simplify the numerator by multiplying using the FOIL method (First, Outer, Inner, Last).
Numerator = (5*5 + 5*(-2i) - 9i*5 - 9i*(-2i))
= (25 - 10i - 45i + 18)
= (43 - 55i)
Step 3: Simplify the denominator using the FOIL method.
Denominator = (5*5 + 5*(-2i) + 2i*5 - 2i*(-2i))
= (25 - 10i + 10i - 4)
= (21)
Step 4: Divide the numerator by the denominator.
Result = (43 - 55i) / 21
Thus, the division of 5-9i by 5+2i is (43 - 55i) / 21.