# Math experts--help with the number "i"

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My son is working in Life of Fred's Advanced Algebra. In his problem sets, he has had these:

i raised to the 4th power = 1

and

i raised to the 10th power = -1

In the text, it is explained that i is in essence the same as square root of -1. So I get the problems above.

But then in his problem set today, he had i raised to the 17th power, and the answer given was i.

Anyone get this? Why wouldn't it be 1 or -1?

TIA

Why wouldn't it be either -1 or 1?

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Every four you get "1." Then you start over with i. So 17 is i. So, 1 x 1 x 1 x 1 x i = i

Edited by Susan C.
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Hello!

I will give this a shot, then one of our true experts can make sure I didn't mess it up!

You are correct: the square root of -1 is i. So, i x i = -1; thus, i cubed is -1 x i = -i, and i to the 4th is -1 x i = 1, since this is the same as i x i x i x i = -1 x -1 = 1. If you find i to the 5th, you start this cycle over again: 1 x i = i. i to the 6th = -1, i to the 7th = -i, i to the 8th = 1, and so on.

So, we can deduce that i to the 16th will be 1, and thus i to the 17th will be 1 x i = i.

Hope this helps!

Blessings,

April

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Below, ^ means "raised to"

i^17 = i^4 * i^4* i^4* i^4* i^1

= 1 * 1 * 1 * i

= i

i raised to any odd power will be i or -i

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Yup to above.

This is also a very good place to see exponential rules in use (although it can be tricky for many students).

Since i^4 is 1, divide 17 by 4.

17 = 4*4+1

So i^17 = i^(4*4+1)

Since a^(m+n) = a^m * a^n, i^17 = i^(4*4) * i^1

Now, since a^(m*n) = (a^m)^n, we can write (i^4)^4 * i^1

Simplifying: i^4=1, so we have 1^4 * i^1 = i.

This would be how you could show what i raised to any power (like 8642) would be. Divide by 4, use exponential rules to rewrite, and you can find the answer in terms of i.

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Yup to above.

This is also a very good place to see exponential rules in use (although it can be tricky for many students).

Since i^4 is 1, divide 17 by 4.

17 = 4*4+1

So i^17 = i^(4*4+1)

Since a^(m+n) = a^m * a^n, i^17 = i^(4*4) * i^1

Now, since a^(m*n) = (a^m)^n, we can write (i^4)^4 * i^1

Simplifying: i^4=1, so we have 1^4 * i^1 = i.

This would be how you could show what i raised to any power (like 8642) would be. Divide by 4, use exponential rules to rewrite, and you can find the answer in terms of i.

Oh dear Lord. My poor kids will SO be on their own when they hit this stage. I am SOOOO lost with this stuff.

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Oh dear Lord. My poor kids will SO be on their own when they hit this stage. I am SOOOO lost with this stuff.

Math is the type of subject that you really really need the foundation solid to move on. Otherwise there's a lot of magical hand-waving and "ta da - the answer!" but with no reason behind WHY the answer is what it is.

I think the whys are what are interesting about algebra.

Here's another explanation (even using the same example).

Exponential rules pop up over and over once you've started using them. This is one of the places where the exponential rules give the "why".

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