Write each power of i as i, - 1, - i, or 1. i 44

Welcome back. I'm so glad you're here. We've got this gorgeous number here. I to the 88th is our expression and we are to write it as negative. 11 negative I or I. Well how do we do that? We recall from previous lessons that when we have I to the first that equals I. When we have I to the second that equals negative one. When we have I to the third that equals negative I and I to the fourth equals positive one. Well those are our four answer choices. But how do we get from 88 to any of those four choices? Well, the key here is this I to the fourth equals one. We want to know how many ones are in I to the 88th. Or in other words how many I to the fourths are in I to the 88th because one times however many ones it is. That's just going to be one. So let's find out how many I to the fourths are an eye to the 88th by dividing 88 by four. While we plug that into a calculator and 88 divided by four is 22. Alright, well there are 22 I. to the fourths and I to the 88 Or in other words we've got I to the fourth raised to the 22nd and I to the fourth again, that's just one. So one raise to the 22nd and one times however many ones it's going to be. That's just one. So our answer here is answer choice. B Well done. We'll catch you on the next one.

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2:01 minutes

Problem 55a

Textbook Question

Write each power of i as i, - 1, - i, or 1.
i⁴⁴

Verified step by step guidance

Step 1: Recall the powers of i follow a repeating cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers of i.

Step 2: To determine the value of i^44, divide the exponent (44) by 4 to find the remainder. This will help identify where in the cycle the power falls.

Step 3: Perform the division: 44 ÷ 4 = 11 with a remainder of 0. The remainder determines the position in the cycle.

Step 4: Since the remainder is 0, i^44 corresponds to i^4 in the cycle. From the cycle, we know that i^4 = 1.

Step 5: Conclude that i^44 simplifies to 1 based on the repeating cycle of powers of i.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Powers of i

The imaginary unit i is defined as the square root of -1. Its powers cycle through four distinct values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cyclical pattern repeats every four powers, which is crucial for simplifying higher powers of i.

Modulus Operation

To simplify powers of i, we can use the modulus operation with respect to 4, since the powers of i repeat every four terms. For example, to find i^44, we calculate 44 mod 4, which equals 0. This indicates that i^44 corresponds to i^0, which is equal to 1.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. Understanding complex numbers is essential for working with i, as it allows for the manipulation and interpretation of expressions involving imaginary units in various mathematical contexts.

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