LanaTron Posted January 6, 2011 Share Posted January 6, 2011 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? Quote Link to comment Share on other sites More sharing options...
Susan C. Posted January 6, 2011 Share Posted January 6, 2011 (edited) 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 January 6, 2011 by Susan C. more Quote Link to comment Share on other sites More sharing options...
April in CA Posted January 6, 2011 Share Posted January 6, 2011 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 Quote Link to comment Share on other sites More sharing options...
mirth Posted January 6, 2011 Share Posted January 6, 2011 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 Quote Link to comment Share on other sites More sharing options...
Dana Posted January 6, 2011 Share Posted January 6, 2011 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. Quote Link to comment Share on other sites More sharing options...
LanaTron Posted January 7, 2011 Author Share Posted January 7, 2011 . Quote Link to comment Share on other sites More sharing options...
cin Posted January 7, 2011 Share Posted January 7, 2011 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. Quote Link to comment Share on other sites More sharing options...
Dana Posted January 7, 2011 Share Posted January 7, 2011 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". Quote Link to comment Share on other sites More sharing options...
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