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Chapter 2: Problem 49

In Exercises 43 to 50 , evaluate the power of \(i .\) $$i^{-34}$$

Short Answer

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Step by step solution

01

Apply the Negative Exponent Rule

The Negative Exponent Rule states that \(a^{-n} = \frac{1}{a^{n}}\), where \(a = i\) and \(n = 34\) in the given scenario. Therefore, \(i^{-34} = \frac{1}{i^{34}}\).
02

Simplify Using the Power Rule

The power of \(i\) follows a cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then returns back to \(i\) for \(i^5\). As \(i^4 = 1\), any power multiple of 4 reduces to 1. Simplifying the exponent 34 by finding the remainder when it is divided by 4 results in 2. Hence, \(i^{34} = i^2 = -1\). Now substitute \(i^{34}\) with -1 in the result from Step 1.
03

Evaluation

Substituting \(i^{34} = -1\) in the equation obtained from Step 1, we get \(\frac{1}{i^{34}} = \frac{1}{-1} = -1\). Therefore, the solution to \(i^{-34}\) is -1.

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Key Concepts

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

Negative Exponent Rule
The Negative Exponent Rule is a crucial concept in mathematics. This rule states that when you have a negative exponent, you can transform it into a positive one by taking the reciprocal. In mathematical terms, this appears as:
  • For any non-zero number \(a\) and any positive number \(n\), \(a^{-n} = \frac{1}{a^n}\).
In the context of complex numbers, this rule works the same way. Consider the problem of evaluating \(i^{-34}\). Here, \(i\) is the base and \(-34\) is the exponent. By applying the negative exponent rule, \(i^{-34}\) is rewritten as \(\frac{1}{i^{34}}\). This simplifies the complexity of handling negative exponents and provides a direct path to work with positives.
Powers of i
The imaginary unit \(i\) has a fascinating property; it cycles through four predictable powers. By understanding this cycle, simplifying expressions with powers of \(i\) becomes straightforward. The four primary results are:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
After \(i^4\), the powers repeat. Therefore, for any integer exponent, you only need to determine the remainder modulo 4. For \(i^{34}\), dividing 34 by 4 leaves a remainder of 2. Thus, \(i^{34} = i^2 = -1\). This cycle simplifies calculations significantly, especially with large exponents.
Imaginary Unit i
The imaginary unit, known as \(i\), is fundamental to the field of complex numbers. It is defined as the square root of \(-1\), which cannot be solved within the realm of real numbers. Therefore, \(i\) allows for the extension into the complex plane. We define it as:
  • \(i^2 = -1\)
This single property lays the foundation for constructing all complex numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers. While working with \(i\), understanding its predictable power cycle is essential to simplify and solve complex expressions efficiently. This profound concept enriches mathematical computations and enables solutions to equations previously unsolvable with only real numbers.

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