Question

Let n be a natural number. Which of the following equals i^4n? a) -1 b) i c) 1 d) -1
If somebody could walk me through this, I will award a medal!

Answer 1

do you mean
\[ i^{4n} \]
?
if so , write it as
\[ \left(i^4\right)^n \]

Answer 2

that uses the rule that
\[ \left(a^b\right)^c = a^{bc}\]
in reverse.

Answer 3

now figure out i^4
i*i * i*i

i*i is what ?

Answer 4

Okay, I think I understand now but I'd really appreciate someone clarifying for me. Since the exponents of i circle around between i, -i, 1, -1 it just depends on the multiple? So i^4n is the same as i^4, which is equal to 1.
Yes sorry for the confusion, I mean \[(i^4)^n\]

Answer 5

i^4 is 1
so your problem is
1^n
where n is a natural number (an integer 1 or bigger)
1^n is 1*1*1*...*1 for n ones
obviously that is just 1

Answer 6

Here, I don't think I wrote it out quite right - it's really meaning i^(4n)
Which I can't seem to make open study write correctly.
Maybe that means the same thing, because the answer to both is 1.

Answer 7

notice we can't play the same trick for i^(3n)
writing it as
(i^3)^n

i*i*i is -i
and
(-i)^n is (-1)^n * i^n
and unless we know more about n, we can't do anything.

Answer 8

Do you know about this rule of exponents:
\[ \left(a^b\right)^c = a^{bc} \]
that is what lets you rewrite i^(4n) as (i^4)^n

Answer 9

Thank you, that explanation was really helpful. It is 1, but not for the reason I thought (: thanks for the refresher!