OPERATIONS ON COMPLEX NUMBERS

3i(2- 8i)=- 6i+24(i^2)=- 6i+24(- 1)=- 6i- 24=- 24- 6i

\sqrt{- 9} \sqrt{- 36} = \sqrt{- 1 \times 9} \sqrt{- 1 \times 36} = \sqrt{- 1} \sqrt{9} \times \sqrt{- 1} \sqrt{36} = i 3 \times i 6 = 18 i^{2} = - 18

(3 - 2i)(1 + 5i) = 3 + 15i - 2i - 10i^{2} = 3 + 13i - 10(- 1) = 13 + 13i

(5 - i)(9 - 3i) = 45 - 15i - 9i + 3i^{2} = 45 - 24i + 3(- 1) = 42 - 24i

Powers of $i$

We now want to simplify certain expressions involving higher powers of $i$ . simplifying powers of $i$ can then be done by using the fact that $i^{2} = - 1$ and expressing the given power of $i$ in terms of $i^{2}$ . Consider:

The powers of $i$ cycle themselves through the values of $i, - 1, - i$ and 1.

Example: Simplify:

a. $i^{11}$ b. $i^{22}$ c. $i^{61}$ d. $i^{100}$

Solution:

i^{11} = (i^{2})^{5}i = (-1)^{5}i = (-1)i = -i

i^{22} = (i^{2})^{11} = (-1)^{11} = -1

Operations on Complex Numbers and Powers of i

Operations on Complex Numbers

Powers of iii

Solution:

Powers of $i$