The next topic that we desire to discuss here is powers of i. Let's just take a look at what occurring while we start looking at many powers of i.
i^{1} = i i^{1} = i
i^{2} = -1 i^{2} = -1
i^{3} =i ⋅ i^{2} = -i i^{3 } = -i
i^{4} = (i^{2} )^{2} = ( -1)^{2} = 1 i^{4} = 1
i^{5} = i ⋅ i^{4} = i i^{5 } = i
i^{6 }= i^{2} ⋅ i^{4} = ( -1) (1) = -1 i^{6 } =-1
i^{7} = i ⋅ i^{6} = -i i^{7} = -i
i^{8 } = (i^{4} )^{2} = (1)^{2} = 1 i^{8 } = 1
Can you notice the pattern? All powers if i can be reduced down to one of four probable answers and they repeat every four powers continuously. It can be a convenient fact to remember.
Next we need to address an issue on dealing along with square roots of -ve numbers. we know that we can do the following.
In other terms, we can break up products within a square root into a product of square roots provided both numbers are positive.
It turns out that actually we can do the same thing if one of the numbers is -ve. For instance,
However, if both of numbers are negative it won't work anymore as the following shows.
We can summarize it as a set of rules. If a & b are both positive numbers then,
Consider the following example to know it's important.