- To figure out the value of j−^3 using the difference of powers law, note that
j−^3 =j^1 −^4=j/j^4
We have determined that j^4 = 1. Therefore,j/j^4 =j/1=j
We can conclude that j−^3 =j. Now let’s determine j−^5. Again using the difference of pow-
ers law, we can say thatj−^5 =j−^1 −^4=j−^1 /j^4
We have found that j−^1 =−j, and also that j^4 = 1. Thereforej−^1 /j^4 = (−j)/1=−j
Now we know that j−^5 =−j. Finally, let’s figure out the value of j−^7. Once again choosing
numbers and applying the difference of powers law, we can say thatj−^7 =j−^3 −^4=j−^3 /j^4
We have found that j−^3 =j, and also that j^4 = 1. Thereforej−^3 /j^4 =j/1=j
This tells us that j−^7 =j. By now, it is apparent that we’ll alternate between −j and j as we
raisej to ever-decreasing negative odd integer powers of −9,−11,−13, and so on.- Refer to Table C-1. The four-way cycle of values goes on forever in both directions, that
is, for positive and negative integer powers of j. - The answers, along with explanations, are as follows.
(a) To find the sum (4 +j5)+ (3 −j8), we add the real parts and the imaginary parts
separately. This gives us(4+j5)+ (3 −j8)= (4 + 3) +j(5− 8)= 7 +j(−3)
= 7 −j 3
Chapter 21 665