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Chapter 9: Problem 80

Simplify. \(i^{43}\)

Short Answer

Expert verified
\(i^{43} = -i\)

Step by step solution

01

Understanding Powers of i

The imaginary unit i follows a cyclical pattern with its powers: \(i = \sqrt{-1}\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). These results repeat every four powers.
02

Determine the Position within the Cycle

To simplify \(i^{43}\), find the remainder when 43 is divided by 4 (since the pattern repeats every 4 powers). \(43 \div 4 = 10\) with a remainder of 3. Thus, \(i^{43}\) is equivalent to \(i^{3}\).
03

Simplify Using the Pattern

Using the known cyclic pattern, \(i^3 = -i\). Therefore, \(i^{43} = -i\).

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Key Concepts

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

Powers of i
Imaginary numbers are a fascinating part of mathematics, especially the imaginary unit, denoted as i. The imaginary unit is defined as the square root of -1, that is, \(i = \sqrt{-1}\). This seemingly simple definition leads to a sequence of powers with interesting properties.

The powers of i follow a repetitive or cyclic pattern:
  • \(i = \sqrt{-1}\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
Notice that after the fourth power, the sequence starts again:

\[ \begin{align*} i^5 &= i \ i^6 &= -1 \ i^7 &= -i \ i^8 &= 1 \end{align*} \] Understanding this cycle is essential because it means we can simplify any power of i by recognizing this repeating pattern.
Cyclic Pattern
As mentioned earlier, the powers of i repeat every four powers, creating a cyclic pattern. This pattern greatly simplifies the process of dealing with higher powers of i.

For example, let's consider a large power such as \(i^{43}\). Without the cyclic pattern, calculating this directly would be tedious. However, recognizing that every fourth power is the same as one of the first four powers simplifies this task immensely.

The cyclic pattern can be summarized as follows:
  • \(i^{4k} = 1\)
  • \(i^{4k+1} = i\)
  • \(i^{4k+2} = -1\)
  • \(i^{4k+3} = -i\)
Here, k is a whole number. This repetition every four powers is what makes calculations with i manageable and simple.
Remainder
To use the cyclic pattern efficiently, particularly with large exponents, we can utilize the concept of the remainder when dividing by 4. Given an exponent, finding how many complete cycles of 4 fit in, and what remains, can provide the equivalent power in the first cycle.

Let's take \(i^{43}\) as an example:
  • First, divide the exponent by 4: \(43 \div 4 = 10\) with a remainder of 3.
  • This tells us \(i^{43}\) is equivalent to \(i^3\) because 43 is 10 full cycles of 4 plus an extra 3.
Using the cyclic pattern where \(i^3 = -i\), we can easily conclude \(i^{43} = -i\).

This remainder approach saves time and makes simplifying powers of i straightforward. Whenever you face such an exponent, just find the remainder when dividing by 4, and refer back to the initial cycle of powers of i.

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Most popular questions from this chapter

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