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Chapter 2: Problem 114

Simplify \(i^{83}\)

Short Answer

Expert verified
The simplified form of \(i^{83}\) is \(-i\).

Step by step solution

01

- Understand Powers of i

Recall that the imaginary unit, represented by the symbol \(i\), has the property that \(i^2 = -1\). From this property, we also have that \(i^3 = i^2 \times i = -1 \times i = -i\) and \(i^4 = i^3 \times i = -i \times i = -i^2 = -(-1) = 1\). After \(i^4\), the powers of \(i\) start repeating: \(i^5 = i\), \(i^6 = -1\), \(i^7 = -i\), \(i^8 = 1\), and so on.
02

- Determine the Equivalent Power Modulo 4

To simplify higher powers of \(i\), calculate the exponent modulo 4. For \(i^{83}\), reduce the exponent 83 modulo 4. Thus, \(83 \mod 4 = 3\). This means that \(i^{83}\) has the same value as \(i^3\).
03

- Simplify Using the Known Cycle

From Step 1, it is known that \(i^3 = -i\). Therefore, \(i^{83} = i^3 = -i\).

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Key Concepts

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

Imaginary Unit
The imaginary unit, symbolized as \( i \), is a fundamental concept in complex numbers. Its primary property is that when squared, it equals \( -1 \). Mathematically, this is expressed as \( i^2 = -1 \). This characteristic sets up a repeating pattern for the powers of \( i \):
  • \( i^3 = i^2 \times i = -1 \times i = -i \)
  • \( i^4 = i^3 \times i = -i \times i = -i^2 = -(-1) = 1 \)
Understanding these basic powers helps in managing and simplifying expressions involving higher powers of \( i \).
Exponent Modulo
When dealing with high exponents of \( i \), it is essential to leverage the repeating cycle of \( i \)鈥檚 powers. Notice that every four powers, the sequence starts anew:
  • \( i^1 = i \)
  • \( i^2 = -1 \)
  • \( i^3 = -i \)
  • \( i^4 = 1 \)
To simplify high powers, use the concept of modulo operation. Modulo divides the exponent by 4 and takes the remainder: \( 83 \mod 4 = 3 \) explains this. Hence, \( i^{83} \) simplifies to \( i^3 \). This efficient method reduces complexity and avoids cumbersome calculations.
Simplification of Complex Numbers
Simplification of complex numbers, especially involving the imaginary unit \( i \), revolves around recognizing patterns and applying basic properties. From the power cycle we know, simplifying \( i^3 \) is straightforward as: \( i^3 = -i \). Consequently, \( i^{83} = -i \). The ability to break down complex expressions into basic components like \( i^3 \) helps in simplifying and solving more complex problems. Remember:
  • Identify the power cycle
  • Use modulo operation
  • Substitute and simplify
    • This structured process assures accuracy and efficiency in working with imaginary numbers.

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Most popular questions from this chapter

Find the slope of the line \(3 x-9 y=7\).

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{3}-5 x+1$$

Use transformations to graph each function and state the domain and range $$y=3 x-40$$

Graph each pair of functions (without simplifying the second function) on the same screen of a graphing calculator and explain what each exercise illustrates. a. \(y=x^{4}-x^{2}, y=(-x)^{4}-(-x)^{2}\) b. \(y=x^{3}-x, y=(-x)^{3}-(-x)\) c. \(y=x^{4}-x^{2}, y=(x+1)^{4}-(x+1)^{2}\) d. \(y=x^{3}-x, y=(x-2)^{3}-(x-2)+3\)

Find the domain and range for \(f(x)=-5 \sqrt{x-3}+2\).
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