Problem 90 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$i^{27}$$

Step-by-Step Solution

Answer

i^27 = -i.

Step 1 - Identify the pattern of powers of i

The powers of the imaginary unit i follow a repeating pattern: i^1 = i i^2 = -1 i^3 = -i i^4 = 1 After i^4, the pattern repeats. Thus, for any positive integer n, the value of i^n can be found by considering the remainder of n divided by 4.

Step 2 - Determine the remainder

Find the remainder when 27 is divided by 4: 27 ÷ 4 = 6 remainder 3. This means 27 can be written as 4*6 + 3.

Step 3 - Use the remainder to simplify

Use the remainder to determine the simplified form of i^27: Since the remainder is 3, i^27 = i^3.

Step 4 - Evaluate i^3

From the pattern identified: i^3 = -i.

Key Concepts

imaginary unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This is a unique number because in the real number system, no number squared gives a negative result.
Understanding the imaginary unit is crucial because it allows us to extend the real numbers to complex numbers. Complex numbers are essential in many fields, including engineering, physics, and applied mathematics.
The imaginary unit allows for the creation of complex numbers like \(a + bi\), where \(a\) and \(b\) are real numbers. This expanded set is very useful for solving equations that don't have solutions within the real numbers alone.

pattern of powers

The powers of \(i\) exhibit an interesting repeating pattern. This pattern is:

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

This cycle repeats every four powers. So, \(i^5 = i\), \(i^6 = -1\), \(i^7 = -i\), and \(i^8 = 1\), and this continues indefinitely.
Recognizing this pattern is very important because it allows for the simplification of expressions involving high powers of \(i\). Instead of multiplying many \(i\)'s together, you can use the pattern to find a simplified result quickly.
For example, knowing the pattern helps us simplify \(i^{27}\), which is part of the given exercise.

remainder division

To simplify higher powers of \(i\), you can use the method of remainder division. This involves dividing the exponent by 4 and using the remainder to determine the result based on the known pattern of powers of \(i\).
For example, let's simplify \(i^{27}\). First, divide 27 by 4. The quotient is 6, and the remainder is 3 (since 27 = 4*6 + 3). The important part here is the remainder.
We can now say that \(i^{27} = i^3\) because the powers of \(i\) repeat every 4. So, the problem now is to find \(i^3\), which is much simpler.

simplification of expressions

Once you have found the remainder in the division, simplifying the expression becomes straightforward.Using our previous example, we determined that \(i^{27} = i^3\). From the pattern of powers, we know that:

• \(i^3 = -i\)

Thus, \(i^{27} = -i\). By using these steps, we turned a seemingly complex problem into a simple one, utilizing the known properties and patterns of the imaginary unit \(i\).
This method can be applied to any power of \(i\), ensuring that expressions involving \(i\) are always manageable. Simplification is a crucial skill, not just for imaginary numbers, but for all areas of mathematics, as it allows for more straightforward and comprehensible solutions.