SOLUTION: We use the Greek letter iota, i, to represent the square root of -1 ( ). We call i an imaginary unit. Any number that is a product of a real number and the imaginary unit i is call

Algebra ->  Expressions-with-variables -> SOLUTION: We use the Greek letter iota, i, to represent the square root of -1 ( ). We call i an imaginary unit. Any number that is a product of a real number and the imaginary unit i is call      Log On


   

Question 936945: We use the Greek letter iota, i, to represent the square root of -1 ( ). We call i an imaginary unit. Any number that is a product of a real number and the imaginary unit i is called an imaginary number. Complex numbers are numbers that consist of a real part and an imaginary part, and they are generally expressed as a + bi. In this task, you will find a pattern relating to the powers of i and also discover some identities involving complex numbers.
a. Use the identity i2 = -1 to compute the powers of i and complete the table.
Type your response here:
Power of i Result
i^3
i^4
i^5
i^6
i^7
i^4n
i^4n+1
i^4n+2
i^4n+3
b. Examine the pattern in the powers of i you wrote in the table, and create a rule for finding the value of large powers of i. Justify your answer.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
a. Use the identity i%5E2+=+-1%29%29%29+to+compute+the+powers+of+%7B%7B%7Bi and complete the table.
Type your response here:
Power of i Result
i%5E3=i%5E2%2Ai=%28-1%29%2Ai=-i
i%5E4=i%5E2%2Ai%5E2=%28-1%29%2A%28-1%29=1
i%5E5=i%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2Ai=i
i%5E6=i%5E2%2Ai%5E2%2Ai%5E2=%28-1%29%2A%28-1%29%2A%28-1%29=-1
i%5E7=i%5E2%2Ai%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2A%28-1%29%2Ai=-i
i%5E4n=%28i%5E4%29%5En=%28i%5E2%2Ai%5E2%29%5En=%28%28-1%29%2A%28-1%29%29%5En=1%5En=1
i%5E%284n%2B1%29=%28i%5E4%29%5En%2Ai=1%2Ai=i
i%5E%284n%2B2%29=%28i%5E4%29%5En%2Ai%5E2=1%2A%28-1%29=-1
i%5E%284n%2B3%29=%28i%5E4%29%5En%2Ai%5E2%2Ai=1%2A%28-1%29%2Ai=-i

b. Examine the pattern in the powers of i you wrote in the table, and create a rule for finding the value of large powers of i. Justify your answer.
b%5En is the product of multiplying n bases:
b%5En=b%2Ab%2Ab.......b+ and b is multiplied by b +n times
so,we can apply same rule if b=+i%5E2 and we will have
%28i%5E2%29%5En=i%5E%282n%29=i%5E2%2Ai%5E2.......i%5E2+...... n times