Problem 95 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$\frac{1}{i^{-11}}$$

Step-by-Step Solution

Answer

-i

Step 1: Understanding the problem

Given the expression \(\frac{1}{i^{-11}}\), the goal is to simplify it by using the properties of the imaginary unit \(i\).

Step 2: Simplify the negative exponent

The negative exponent can be simplified as \(\frac{1}{i^{-11}} = i^{11}\). This is because \(\frac{1}{a^{-b}} = a^{b}\). So \(i^{-11}\) becomes \(i^{11}\) when moved to the numerator.

Step 3: Apply the powers of i

Recall the cyclic nature of powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Since the powers of \(i\) repeat every 4, reduce the exponent: \(i^{11} = i^(11 \mod 4) = i^3\).

Step 4: Simplify final result

Using the value from the powers of \(i\), \(i^3 = -i\). Therefore, \(\frac{1}{i^{-11}}\) simplifies to \(-i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is denoted by the symbol \(i\). It is defined as the square root of negative one, meaning \(i^2 = -1\). The concept of imaginary numbers extends the real number system \(\textbf{R}\) to the complex number system \(\textbf{C}\), which includes all numbers of the form \(a + bi\) where \(a\) and \(b\) are real numbers.

Understanding the properties of \(i\) is crucial. The significance of \(i\) lies in its ability to represent numbers that are not possible within the set of real numbers. For example:

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

Notice how the powers of \(i\) exhibit a cyclic nature, which is another important property.

Negative Exponent

A negative exponent indicates that the base is on the wrong side of a fraction line and needs to be inverted to become positive. Specifically, \(a^{-b} = \frac{1}{a^b}\). For the given exercise of simplifying \(\frac{1}{i^{-11}}\):

• The negative exponent \(i^{-11}\) translates to \(\frac{1}{i^{-11}} = i^11\) when moved to the numerator using the property \(\frac{1}{a^{-b}} = a^b\).

By handling negative exponents properly, we convert seemingly complex expressions into simpler forms that can be further worked with.

Cyclic Nature of Powers of i

The powers of \(i\) exhibit a cyclic behavior, repeating every four exponents:

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

This cycle helps to simplify powers of \(i\) efficiently.
To simplify \(i^{11}\):

• Find the remainder when 11 is divided by 4, which is 3 (since 11 mod 4 is 3).
• Therefore, \(i^{11} = i^3\).
• From the cycle, \(i^3 = -i\).

Thus, using the cyclic nature of \(i\), we conclude that \(\frac{1}{i^{-11}}\) simplifies to \(-i\).
This property is particularly useful in simplifying complex expressions involving high powers of \(i\).