Simplify the power of i i i . i 1003 i^{1003} i 1003

Hey, y'all. Let's take a look at this practice problem. So here I have I to the power of 1,003, and I wanna simplify this power of I. So I'm gonna ask myself, is this 1,003 evenly divisible by 4? And my answer is no, so that means I need to find my remainder of dividing by 4. So let's go ahead and take 1,003 and divide it by 4 using long division. So 4 doesn't go into 1 at all, but it does go into 10 two times. So I have that 8. 10 minuteus 8 will give me 2, and then I have 20. 4 goes into 20 exactly 5 times. So if I take 20 20, don't have anything there. I need to pull my 3 down. And 4 cannot go into 3 at all, so it goes in 0 times. And I am simply left with 3 as my remainder. So that means that I to the power of 1,003 is really just I cubed. So our remainder here is 3 I cubed, which just gives me negative I. So I to the 1,003rd power is just negative I. Thanks for watching, and I'll see you in the next one.

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1. Equations and Inequalities

Powers of i

Precalculus

1. Equations and inequalities

Powers of i

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Multiple Choice

Simplify the power of $i$ .
$i^{1003}$

$i$

$-1$

$-i$

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Verified step by step guidance

First, understand the properties of the imaginary unit, denoted as $ i $. The imaginary unit $ i $ is defined such that $ i^2 = -1 $.

Next, recognize the cyclical nature of powers of $ i $. Specifically, $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, and $ i^4 = 1 $. This cycle repeats every four powers.

To simplify $ i^{1003} $, determine the remainder when 1003 is divided by 4, since the powers of $ i $ repeat every 4 steps.

Calculate $ 1003 \div 4 $ to find the remainder. The remainder will tell you which power in the cycle $ i^1, i^2, i^3, i^4 $ corresponds to $ i^{1003} $.

Use the remainder to identify the equivalent power of $ i $. If the remainder is 1, $ i^{1003} = i $; if 2, $ i^{1003} = -1 $; if 3, $ i^{1003} = -i $; and if 0, $ i^{1003} = 1 $.