/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Find each power of \(i\) $$ ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 77

Find each power of \(i\) $$ i^{89} $$

Short Answer

Expert verified
\(i^{89} = i\)

Step by step solution

01

- Understand the Powers of i

First, recall the fundamental powers of the imaginary unit, i. They follow a repeating cycle every four powers: \(i^1 = i \) \(i^2 = -1 \) \(i^3 = -i \) \(i^4 = 1\) This pattern repeats every four exponents.
02

- Find the Remainder

To determine which part of the cycle the given power falls into, calculate the remainder of 89 divided by 4.\(89 梅 4 = 22\text{ R }1 \)Thus, the remainder is 1.
03

- Find the Equivalent Power

Since the remainder is 1, the given power \(i^{89}\) is equivalent to \(i^1\).
04

- Determine the Value

Finally, use the basic power's cycle determined in Step 1:\(i^1 = i\)Therefore, \(i^{89} = i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

imaginary numbers
Imaginary numbers are a fundamental component of complex numbers. They are used to extend the real number system and allow us to solve equations that do not have real solutions, such as \(x^2 = -1\). The symbol \(i\) is used to represent the imaginary unit, with the key property \(i^2 = -1\). This makes it possible to work with square roots of negative numbers in a consistent way.

In dealing with imaginary numbers, it's important to remember:
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Most popular questions from this chapter

Concept Check Which sum could be simplified without first simplifying the individual radical expressions? A. \(\sqrt{50}+\sqrt{32}\) B. \(3 \sqrt{6}+9 \sqrt{6} \quad\) C. \(\sqrt[3]{32}+\sqrt[3]{108}\) D. \(\sqrt[5]{6}+\sqrt[5]{192}\)

Multiply. $$ (2-i)^{2}(2+i)^{2} $$

Multiply. $$ (1+i)^{2}(1-i)^{2} $$

Solve each formula for the indicated variable. $$ r=\sqrt{\frac{M m}{F}} \text { for } M $$

Perform the indicated operations. Give answers in standard form. $$ \frac{3}{2-i}+\frac{5}{1+i} $$
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