- #1

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I know the answer , but cannot get my answer to be correct. Can someone help me solve this?

Thanks

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- Thread starter Wasper
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In summary, the conversation revolved around expressing (i-3)^4 in simplest a+bi form and the steps involved in simplifying the expression. The conversation also included helpful tips and strategies for manipulating imaginary numbers.

- #1

- 6

- 0

I know the answer , but cannot get my answer to be correct. Can someone help me solve this?

Thanks

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- #2

Ok, I'd break it down like this;

(i-3)(i-3) = i^2-6i+9

Now, because the original is to the fourth power you would have two groups like this;

(i^2-6i+9)(i^2-6i+9)

Multiply all the above out properly and after combining like terms you have;

= i^4-12i^3+54i^2-108i+81

Now, using what you know about 'i' raised to a power (how it repeats, and so forth) you will be able to transform some i's into numbers. The numbers will then combine and reduce to;

28-96i

Well, at least I think that's about right, haha, because I think I goofed on one of the signs but I'm too lazy to go back over it again and check my work. I think you can take it from here. It's actually quite a fun thing to do when you start getting the hang of it and have your mind free to concentrate.

[edit]

Ooops, I just noticed this was homework help

Anyway, since I've gone this far already and because I don't know your level of understanding I'll rehash what I did above a little;

(i-3)^4 = (i-3)(i-3)(i-3)(i-3)

I broke it down into two groups of;

[(i-3)(i-3)]x[(i-3)(i-3)] = [i^2-6i+9]x[i^2-6i+9]

Do you follow?

You could have just kept grunting them out by only multiplying the quantity (i-3) over and over, but the way I did it seemed more of a shortcut (but it is probably easier to confuse yourself doing it that way too). Lastly, the imaginary part might seem confusing at first but is actually rather easy if you beat it into your brain that because i = squareroot of negative one you get the following repeating patern;

i^1 = sqrt -1

i^2 = (sqrt -1)(sqrt -1) = -1

i^3 = (-1)(sqrt -1) = (-1)(i) = -i

i^4 = (-1)(-1) = 1

Looking at the results from above you can see that i^5 is just going to be (sqrt -1)(1) = sqrt -1 = i^1, so i^5 ends up being i^1 (pattern repeats all over again). Do you suppose the odds are high that i^6 will end up being equivalent to i^2 ?? Yes, because it would be the equivalent of (i^2)(i^4) = (-1)(1) = -1 = i^2.

On i^4, an easy way to determine the answer is is by recalling what i^2 was equal to). You

= (-i)(sqrt -1)

= (-i)(i)

= -i^2

= (-i)(-i)

= (-sqrt -1)(-sqrt -1)

= -[(sqrt -1)(sqrt -1)]

= -[-1]

= 1

There, I hope that makes up for my having blabbed to much.

Last edited by a moderator:

- #3

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(i^2- 6i+ 9)(i^2- 6i+ 9) in that form.

i^2- 6i+ 9 is, of course, -1- 6i+ 9= 8- 6i so it is simpler

to multiply (8- 6i)(8- 6i)= 64- 2(8)(6)i+ 36i^2= 64- 96i- 36

= 28- 96i just as BoulderHead said.

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