Question

How do I simplify i to the power of 23?

Answer 1

Notice this pattern

i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1
...

it keeps going and it repeats on every 4th number (starting from 1). That means i^1 = i^5 for instance or i^7 = i^3

Answer 2

You don't have to keep writing out that pattern to get to i^23, you can use this rule below

Step 1) Take the exponent over 'i' and divide it by 4

Step 2) The remainder you get will tell you what the answer is when you compare it to that pattern.

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So let's use these steps to compute i^23

Step 1) Divide the exponent 23 by 4: 23/4 = 5 remainder 3

Step 2) The remainder is 3. So that means i^23 = i^3 = -i. In short, i^23 = -i

Answer 3

Another example:

Let's say we want to find i^56


Step 1) Divide the exponent of 56 by 4 to get 56/4 = 14 remainder 0

Step 2) The remainder is 0, so i^56 = i^0. Because i^0 = 1, we know that i^56 = 1