Problem 94 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$i^{-14}$$

Step-by-Step Solution

Answer

i^{-14} = -1

Step 1: Understand the Powers of i

Powers of the imaginary unit i follow a repeating pattern every four exponents:i = ii^2 = -1i^3 = -ii^4 = 1i^5 = i (which is the same as i^1)

Step 2: Determine the Remainder

Find the remainder when -14 is divided by 4. This helps in identifying the equivalent positive exponent.i^{-14} i^{-14} = (i^4)^{-3.5}The powers of i repeat every 4, so divide -14 by 4:-14 ÷ 4 = -3 remainder -2This means i^{-14} is equivalent to i^{-2}.

Step 3: Simplify Using Positive Exponents

Convert the negative exponent to a positive one:i^{-2} = 1/(i^2). Since i^2 = -1, we have:1/(-1) = -1.

Key Concepts

imaginary unit

The imaginary unit, denoted as \( i \), is a fundamental concept in mathematics, especially in complex numbers. The imaginary unit is defined by the property that \( i^2 = -1 \). It allows us to extend the real number system to include solutions to equations that otherwise wouldn't have real solutions, such as \( x^2 + 1 = 0 \). In this context, \( i \) is defined so that:\( i = \sqrt{-1} \); meaning when squared, gives -1.
Understanding the behavior of \( i \) especially with its powers makes simplifying complex equations more manageable. For instance, the powers of \( i \) follow a repeating cycle every four exponents, making them easy to remember:

• \( i^1 = i \)
• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \)
• \( i^5 = i \) (which restarts the cycle)

This cyclical nature allows us to handle any power of \( i \) by finding its equivalent exponent within the range of 1 to 4.

remainder

The remainder is what’s left over after performing division. When dealing with powers of \( i \), the remainder helps simplify the exponent by reducing it to one within the repeating cycle (1 to 4).
In the given problem, we need to simplify \( i^{-14} \). To do that, we find the remainder when -14 is divided by 4 because the powers of \( i \) repeat every 4. Performing the division, we get:
-14 ÷ 4 = -3 remainder -2
This tells us that \( i^{-14} \) is equivalent to \( i^{-2} \). By using the modulo operation, we can simplify further:
-2 modulo 4 = 2
This tells us \( i^{-2} \) is equivalent to \( i^2 \).
Hence, understanding and calculating the remainder is crucial for simplifying powers of \( i \).

negative exponent

Negative exponents indicate that we should take the reciprocal of the base raised to the positive version of the exponent. In simpler terms, \( a^{-b} = 1/a^b \).
In the context of imaginary numbers, let's handle \( i^{-2} \) as an example:
Applying the rule of negative exponents:
\( i^{-2} = 1/i^2 \)
From our knowledge of powers of \( i \), we know that \( i^2 = -1 \). Therefore:
\( 1/i^2 = 1/(-1) = -1 \)
By converting negative exponents to positive ones, it becomes easier to understand and simplify expressions involving powers of \( i \). Thus, mastering the concept of negative exponents is essential for dealing with more complex problems in algebra and beyond.