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Chapter 10: Problem 79

Find each power of \(i\) $$ i^{-6} $$

Short Answer

Expert verified
The value of \(i^{-6}\) is \(-1\).

Step by step solution

01

Recall the Powers of i

The numbers based on powers of the imaginary unit \(i\) repeat in cycles of four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). This cycle repeats: \(i^5 = i\), \(i^6 = -1\), etc.
02

Simplify the Negative Exponent

The given expression is \(i^{-6}\). First, use the property of exponents \(a^{-b} = \frac{1}{a^b}\) to rewrite the expression as \(\frac{1}{i^6}\).
03

Determine \(i^6\) Using the Cycle

From the cycle in Step 1, we know that \(i^6 = -1\) because the cycle shows that \(i^2 = -1\) and \(6 \mod 4 = 2\). Hence, \(i^6\) falls at \(i^2\).
04

Simplify the Expression

Substitute the value found in Step 3 into the expression from Step 2: \(\frac{1}{i^6} = \frac{1}{-1} = -1\).

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Key Concepts

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

Imaginary Numbers
Imaginary numbers might sound like they're made-up, but they are a crucial part of algebra that help solve equations involving square roots of negative numbers. The imaginary unit is denoted by the letter \(i\), and is defined such that \(i^2 = -1\). This definition provides a tool for dealing with square roots of negative numbers, which were once thought impossible.

Here's how it breaks down:
  • The basic form: Any imaginary number can be expressed as a multiple of \(i\). For example, \(3i, -5i\).
  • Complex numbers: When combined with a real number, it forms a complex number, such as \(3 + 4i\).
  • Applications: Imaginary numbers, and by extension complex numbers, are useful in various fields such as engineering, quantum physics, and applied mathematics.
Understanding imaginary numbers allows you to expand your knowledge from the real number line to the complex plane, opening the door to even more advanced mathematical concepts.
Exponent Rules
The rules of exponents are a set of fundamental laws that help simplify expressions involving powers. These rules stay consistent, whether you are dealing with whole numbers, fractions, or even imaginary numbers.

The key rules include:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \cdot n}\)
  • Power of a Product: \((ab)^n = a^n \cdot b^n\)
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
  • Zero Exponent: \(a^0 = 1\), provided \(a eq 0\)
When applying these rules to imaginary numbers, such as expressing \(i^{-6}\), it鈥檚 important to simplify the negative exponent by transforming it into a fraction using \(a^{-b} = \frac{1}{a^b}\). This transformation ensures that you can use the cyclic pattern of powers to find the value of any power of \(i\).
Cyclic Patterns
Cyclic patterns offer an efficient way to manage powers of imaginary numbers. Since the powers of \(i\) repeat in a cycle of four, it simplifies otherwise cumbersome calculations.

Here鈥檚 how this cyclicity works:
  • Powers of \(i\): Begin with \(i^1 = i\) and follow the cycle: \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).
  • Repeating Cycle: The cycle repeats every four exponentiations. Thus, \(i^5 = i\) and so on. To find the power, determine the remainder when dividing the exponent by four.
  • Use of Modulus: To find any power of \(i\), simply calculate the exponent modulo 4. This gives the equivalent position in the cycle. For example, \(i^6\) is equivalent to \(i^2\) because \(6 \mod 4 = 2\).
Cyclic patterns not only streamline calculations but also help ensure that your answers remain consistent, particularly in algebraic equations that incorporate imaginary numbers. Recognizing this cycle allows you to find powers of these numbers quickly and accurately.

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Most popular questions from this chapter

Which of the following are not real numbers? $$\sqrt[15]{-17}$$

Use a calculator to write a four-decimal-place approximation of each number. $$ 20^{1 / 5} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

Identify the domain and then graph each function. $$g(x)=\sqrt[3]{x+1}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {g(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline-2 & {} \\ \hline 7 \\ \hline-9 \\ \hline \end{array}$$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$f(0)$$
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