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Powers of i - Concept
To determine the value of i raised to a power greater than two, we rewrite the term using exponent rules.Remember that i^2 = -1 and i^4 = 1. Therefore, any exponent of i that is a multiple of four will equal one; any even exponent not divisible by four will equal negative one. Also, negative exponents indicate a reciprocal of the base; if i is in the denominator, it will need to be rationalized.
Dealing with different powers of i. So we know that i to the first is just the square root of -1 which we typically just leave as i. So this is the square root of -1 or i. We also know that i squared is going to be -1. But we also can have different powers of i. So what I'm going to do is sort of go down the row and talk about different powers.
So i cubed. Easy way to figure out i cubed is the same thing as i squared times i. Remember when we multiply bases we add the exponents. So this is i squared times i to the first, this will be i to the third. i squared we know it should be -1 from right here and i is just i. We multiply -1 and i we end up with -i. Okay.
i to the fourth. By the same logic as we did up here, i to the fourth is just going to be i to the third times i. i to the third is -i and i is just i. So what we have then is -i times i -i squared. i squared is -1 so what we end up is with is negative -1 or just 1. Okay.
i to the fifth. By the same logic we did any of these, i to the fifth is just going to be i to the fourth times i but i to the fourth is just 1. So what we really end up is 1 times i or just i. So what we have here is i to the fifth is the same thing as i. If we multiply it by one more i, It's just going to take us the same thing as i times 1, one more i which is just going to be -1. Multiply this by i, i squared -1, sorry -1.
So what we actually end up with is a cycle. i to the sixth is going to be the same thing as i squared, i to the seventh is going to be the same thing as i cubed and i to the eighth is just going to be the same thing as i to the fourth, which is 1. Okay?
So, what we really have is everything is going to repeat after i to the fourth. Every single multiple of i of 4, is just going to get, i to the multiple of 4 is just going to give us 1. Okay?
So, if you think about it. We know i to the fourth is 1, we know i to the eighth is 1. So what about i to the twelfth? Same exact thing. This is once again going to give us 1. What about i to the four hundredth? Okay. Four hundred is a multiple of 4, it's going to lie on one of these ones. So this is going to be 1. i to the 401. We know that i to the 400 is going to land us right here on the i to the fourth power. So we just have one more i which will just bump us to the fifth, equivalent of i to the fifth which is just going to be i.
So whenever we're dealing with powers of i, it's easier just to sort of draw it in comparison to the powers of 4 which we know and then go off of how many more you need to deal with.