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Chapter 10: Problem 86

Find each power of i. $$ i^{26} $$

Short Answer

Expert verified
\(i^{26} = -1\).

Step by step solution

01

Understand the Powers of i

The imaginary unit \(i\) has a cyclic pattern for its powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This pattern repeats every four powers.
02

Find the Remainder of the Exponent Modulo 4

To simplify \(i^{26}\), determine the remainder when 26 is divided by 4. Calculate \(26 \mod 4\).
03

Perform the Division

When 26 is divided by 4, the quotient is 6 and the remainder is 2. Therefore, \(26 \equiv 2 \mod 4\).
04

Use the Remainder to Determine the Equivalent Power

Since \i^{26}\ has the same remainder as \(i^2\), we use the cyclic pattern established earlier. Therefore, \(i^{26} = i^2 = -1\).

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Key Concepts

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

cyclic pattern
When dealing with powers of the imaginary unit, denoted as \(i\), it's important to understand its cyclic pattern. This pattern is the repetition of the values that powers of \(i\) take on. The pattern goes as follows:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
. After reaching \(i^4 = 1\), the sequence starts over. This means that \(i^5 = i\), \(i^6 = -1\), and so on.Breaking down the powers of \(i\) into this smaller, repeating cycle makes calculating higher powers of \(i\) much easier.
remainder modulo operation
In mathematics, performing a modulo operation involves dividing one number by another and finding the remainder. For simplifying powers of \(i\), we use modulo 4 because of the cyclic pattern. For example, to simplify \(i^{26}\), we need to find the remainder of 26 divided by 4. Symbolically, this is written as \(26 \mod 4\).
When you divide 26 by 4, the quotient is 6 and the remainder is 2. So, \(26 \mod 4 = 2\). This tells us that \(i^{26}\) is equivalent to \(i^2\) based on our cyclic pattern.
simplifying powers of i
To simplify any power of \(i\), follow these steps:
  • Identify the power of \(i\) you need to simplify, e.g., \(i^{26}\).
  • Find the remainder of the exponent when divided by 4 using the modulo operation, e.g., \(26 \mod 4 = 2\).
  • Use the cyclic pattern of \(i\). From our pattern, \(i^2 = -1\). Therefore, \(i^{26}\) simplifies to \(i^2\), which equals \(-1\).
.Using this method ensures you can easily simplify even very large powers of \(i\). This technique not only saves time but also reduces the complexity significantly. So next time you encounter a power of the imaginary unit, remember to use these steps to simplify it quickly and correctly.

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