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Chapter 2: Problem 95

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Short Answer

Expert verified
The statement is false.

Step by step solution

01

Simplify \(i^{44}\)

Since 44 is divisible by 4, \(i^{44}\) simplifies to 1. So, \(i^{44} = 1\).
02

Simplify \(i^{150}\)

150 divided by 4 gives a remainder of 2. Therefore, \(i^{150}\) simplifies to \(-1\). So, \(i^{150} = -1\).
03

Simplify \(i^{74}\)

74 divided by 4 gives a remainder of 2. Therefore, \(i^{74}\) simplifies to \(-1\). So, \(i^{74} = -1\).
04

Simplify \(i^{109}\)

109 divided by 4 gives a remainder of 1. Therefore, \(i^{109}\) simplifies to \(i\). So, \(i^{109} = i\).
05

Simplify \(i^{61}\)

61 divided by 4 gives a remainder of 1. Therefore, \(i^{61}\) simplifies to \(i\). So, \(i^{61} = i\).
06

Add the simplified expressions

Adding the simplified expressions, we get \(1 - 1 - 1 + i + i = 2i - 1\).
07

Compare with given value

Comparing the calculated value, \(2i -1\), with the given value -1, we find that they are different. Therefore, the statement is false.

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