Problem 85 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$i^{25}$$

Step-by-Step Solution

Answer

\(i^{25} = i\)

Step 1: Understand the Powers of i

Recall that the imaginary unit, represented by i, follows a cyclic pattern in its powers: \[i^1 = i, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1\] After every fourth power, the cycle repeats.

Step 2: Find the Remainder

Determine how 25 fits into the cycle by finding the remainder of 25 when divided by 4. Calculate \(25 \div 4\) which gives a quotient of 6 and a remainder of 1. Therefore, \(25 ≡ 1 \pmod{4}\).

Step 3: Simplify Using the Remainder

From the remainder 1, we determine that \(i^{25}\) is equivalent to \(i^{1}\). Since \(i^{1} = i\), \(i^{25} = i\).

Key Concepts

imaginary unit

In mathematics, the imaginary unit is a fundamental concept in complex numbers. It is denoted by the symbol \(i\). The defining property of \(i\) is that it satisfies the equation \(i^2 = -1\). This seems counterintuitive because there is no real number that squares to a negative number. But in the realm of complex numbers, we embrace this idea.

This imaginary unit allows us to extend the real number system and solve equations that would otherwise have no solutions in real numbers. For example, the equation \(x^2 + 1 = 0\) has no real solutions, but in the complex number system, it has solutions \(x = i\) and \(x = -i\).

Understanding the imaginary unit is crucial because it opens up a wider world of mathematical solutions and applications in various fields such as engineering, physics, and signal processing.

cyclic pattern

The imaginary unit \(i\) follows a cyclic pattern when raised to successive powers. This pattern repeats every four powers as you have seen in the step-by-step solution:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

This cycle then repeats indefinitely: \(i^5 = i\), \(i^6 = -1\), and so on.

Recognizing this pattern makes it much simpler to simplify powers of \(i\). Instead of continually multiplying \(i\), you can identify where you are in the cycle and find the equivalent simpler expression. This is a powerful tool for solving complex number problems efficiently.

modulo operation

The modulo operation is a mathematical operation that finds the remainder after division of one number by another. It is denoted using the symbol \(\%\) or the word 'mod'.

In the context of simplifying powers of \(i\), the modulo operation helps us determine our position within the cyclic pattern. For example, to simplify \(i^{25}\), we can use:

1. Divide 25 by 4:
25 \div 4 = 6 with a remainder of 1.

2. This means \(25 \equiv 1 \pmod{4}\).

This tells us that \(i^{25}\) is in the same position as \(i^1\) in the cycle. Therefore, \(i^{25} = i\).

Using the modulo operation in this way can significantly simplify calculations involving large powers and cyclic patterns, providing quick and reliable results.