This is only for people that know Math.?

Question

I'm not dumb or anything, I just suck at Math. [Seriously, I'd have a 96 average if I didn't get 80 in Math. And the most simplistic things in Math [like imaginery numbers] are hard for me. Who the hell cares about them? THEY'RE IMAGINERY!

Anyways, here's the question.

"Write the given power of i, in simplest terms as 1,i,-1, or -i.
i to the 300th power = ? "

Thaks a bunch =]

Answer 1

I happen to care quite a bit about imaginary numbers as a person who studies electrical engineering. In electrical engineering, we use the imaginary number (denoted i or j) to help represent sinusoidal waveforms, work with linear systems in the frequency domain, solve sinusoidally-excited RLC circuits etc. It shows up quite a bit in my chosen area of study. Even something as mundane as the voltage sine wave at a common household outlet can be represented by a phasor 120*e^(i*theta) where theta is the phase shift of the sinusoid. Also, I hear physicists use imaginary numbers in some applications.

Anyway, one of the interesting properties of the imaginary number is that when you square it you have -1 (since i = sqrt(-1)).

So:
i^300 = (i^2)(i^2)(i^2)...(i^2)
i^300 = (-1)(-1)(-1)...(-1)

The above set of (i^2) factors will repeat 150 times (the exponents add). Since (-1)^150 = 1 we have our answer (check with your calculator or realize that an even amount of -1 factors will always give you 1).

i^300 = 1

Answer 2

1

Answer 3

Here's how i works. A square root is written as 1/2 power. Anything to the 1/2 power times that thing to the 1/2 power is the original value. Like 3^1/2 times 3^1/2 is 3. So we
can say that the square of the square root of 3 is 3.
There's no mystery to that. It's because the square root is the
inverse of the square .
Now i is a square root of -1. That is, it's a 1/2 power of -1 . Recall, the square of a square root is the original number. So the square of the square root of -1 is -1. But the square root of -1 is defined as i. So we write mathematically
i^2=-1. Now i^3=(i)i^2=(i)(-1)=-i .
Continuing, i^4=(i^2)(i^2)=(-1)(-1)=1. So
i^5=i(i^4)=i(1)=i. Do you see that when we hit the power of 4,
the result was 1 and we start all over again as we increase the
power. i=i^5=i^9=i^13 ... (the powers are 4mod 1)
i^2 = i^6 = i^10 ... (the powers are 4 mod 2)
i^3 = i^7 = i^11 ... (the powers are 4 mod 3)
i^4 = i^8 = i^12 ... (the powers are 4 mod 0)

Now take i^300. 300 is 4 mod 0 so it's equal to i^4 which is 1.

Ta-daa . i^300=1

Answer 4

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Answer 5

i^2= -1
i^4 = (i^2)^2 = (-1)^2 = 1
i^6 = (i^4)*(i^2) = 1 * -1 = -1
i^8 = (i^6)*(i^2) = -1 * -1 = 1
etc.

So, when the power is even, if it is a multiple of 4, the answer's 1. When it's even but not a multiple of 4, the answer's -1.

300 is a multiple of 4, so i^300 = 1

Answer 6

i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
...

since you can combine multiplied terms with the same base by adding the powers of numbers...

i^(4)* = 1
i^(4) * i^(4) = 1 * i^(4)
i^(4+4) = 1 * 1
i^(8) = 1

without going into an actuall "proof" of why this is, i to any multiple of 4 will equal 1.

4 * 75 = 300, so

i^(300) = 1

so for kicks, what is i^(303)?

i^(303) = i^(300 + 3) = i^(300) * i^(3) = 1 * (-i) = -i

I hope I did that right. ;-P

Answer 7

i to the 300th power = i^(4*75)
but i^4 =1 therefore the answer is 1.

Answer 8

Nwordjim gives the key to it: it's according to odd or even powers.

Seriously, I have a minor in math and I love math, but I dropped out of Electrical Engineering school because EVERYTHING is based on imaginary numbers!!! I like REAL numbers! I wound up getting an MBA-Accounting! LOL

Answer 9

i^{300} = 1 !