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Chapter 7: Problem 90

Simplify. $$i^{17}$$

Short Answer

Expert verified
i^{17} = i

Step by step solution

01

- Understand the properties of imaginary unit i

First, recall that the imaginary unit is defined as the square root of -1. The powers of the imaginary unit follow a specific cycle: \[ i^1 = i \]\[ i^2 = -1 \]\[ i^3 = -i \]\[ i^4 = 1 \]The pattern repeats every four powers.
02

- Determine the remainder when dividing the exponent by 4

To simplify \(i^{17}\), find the remainder when 17 is divided by 4 because the powers of \(i\) repeat every 4 steps. \[ 17 \, / \, 4 = 4 \, R1 \]This means \( 17 \) divided by 4 gives a quotient of 4 and a remainder of 1.
03

- Apply the cycle pattern using the remainder

Using the remainder found in Step 2, the power can be reduced to the equivalent lower power. Since the remainder is 1, \[ i^{17} = i^1 = i \]

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Key Concepts

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

Powers of Imaginary Unit
Imaginary numbers often confuse students, but with some practice, you'll find that they follow simple rules. The imaginary unit is defined by the equation \(i = \sqrt{-1}\). It's crucial to understand what happens when you raise \(i\) to different powers. Interestingly, the powers of \(i\) repeat in a cycle of four.

Here is the pattern this creates:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

Because this pattern repeats every four powers, you only need to know these four cases to simplify any power of \(i\).
For instance, if you have \(i^{17}\), you'll eventually find that \(i^{17} = i^1\) by using this repetitive cycle.
Remainder Theorem
To simplify powers of \(i\) like \(i^{17}\), we use something called the Remainder Theorem. The theorem helps in determining how far along the cycle we are by finding the remainder of the power when divided by the cycle length, which is 4 for \(i\).

Here鈥檚 how it works:
  • You divide the exponent by 4.
  • The quotient (discarded here) is not important; the remainder tells you which power in the cycle you are at.

For example, dividing 17 by 4 gives a quotient of 4 and a remainder of 1. This operation tells us that \(i^{17}\) is the same as \(i^1\). So, \(i^{17} = i^{1} = i\).
Cycle of Powers
The term 'cycle of powers' refers to how the powers of \(i\) repeat every four steps. Once you recognize the cycle, you can easily simplify any power of \(i\) this way:
  • After every four powers, the values start to repeat.
  • It's like a loop: \(i, -1, -i, 1, i, -1, -i, 1,...\)

This cycling pattern makes it manageable to handle even very high powers of \(i\). For instance, knowing this pattern lets you simplify \(i^{50}\) by noting:

  • 50 divided by 4 is 12 with a remainder of 2.
  • Thus, \(i^{50} = i^2 = -1\).

Remember, once you find the position within one cycle using the remainder, you can easily identify the simplified form of any power of \(i\).

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