/*! 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 31 Simplify: (a) \(i^{0}, i^{3}, i^... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 31

Simplify: (a) \(i^{0}, i^{3}, i^{4},(\mathrm{b}) \quad i^{5}, i^{6}, i^{7}, i^{8},(\mathrm{c}) \quad i^{39}, i^{174}, i^{252}, i^{317}\)

Short Answer

Expert verified
The simplified forms of the given expressions are: (a) \(i^0 = 1\), \(i^3 = -i\), \(i^4 = 1\) (b) \(i^5 = i\), \(i^6 = -1\), \(i^7 = -i\), \(i^8 = 1\) (c) \(i^{39} = -i\), \(i^{174} = -1\), \(i^{252} = 1\), \(i^{317} = i\)

Step by step solution

01

Find the Remainders of the Powers When Divided by 4

We will find the remainders of each power when divided by 4: a) \(i^0: \quad 0 \mod 4 = 0\) \(i^3: \quad 3 \mod 4 = 3\) \(i^4: \quad 4 \mod 4 = 0\) b) \(i^5: \quad 5 \mod 4 = 1\) \(i^6: \quad 6 \mod 4 = 2\) \(i^7: \quad 7 \mod 4 = 3\) \(i^8: \quad 8 \mod 4 = 0\) c) \(i^{39}: \quad 39 \mod 4 = 3\) \(i^{174}: \quad 174 \mod 4 = 2\) \(i^{252}: \quad 252 \mod 4 = 0\) \(i^{317}: \quad 317 \mod 4 = 1\)
02

Use Cyclical Pattern to Simplify

Now that we have the remainder of each power when divided by 4, we will use the cyclical pattern to simplify each expression: a) \(i^0 = 1\), since the remainder is 0. \(i^3 = -i\), since the remainder is 3. \(i^4 = 1\), since the remainder is 0. b) \(i^5 = i\), since the remainder is 1. \(i^6 = -1\), since the remainder is 2. \(i^7 = -i\), since the remainder is 3. \(i^8 = 1\), since the remainder is 0. c) \(i^{39} = -i\), since the remainder is 3. \(i^{174} = -1\), since the remainder is 2. \(i^{252} = 1\), since the remainder is 0. \(i^{317} = i\), since the remainder is 1.

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