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Chapter 8: Problem 27

Simplify each expression to a single complex number. $$ i^{17} $$

Short Answer

Expert verified
The simplified expression is \( i \).

Step by step solution

01

Identify the Pattern of Powers of i

The powers of the imaginary unit \( i \) follow a repeating pattern: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \). This pattern repeats every four powers.
02

Determine the Remainder of the Exponent Divided by 4

To find which part of the cycle the exponent 17 falls into, divide 17 by 4. The remainder tells us which power in the cycle to use. Calculating this division: \( 17 \div 4 = 4 \) remainder \( 1 \). This means \( i^{17} = i^1 \).
03

Apply the Pattern to Simplify the Expression

Since the remainder is 1, \( i^{17} \) corresponds to \( i^1 \). According to the cycle, \( i^1 = i \). Therefore, the simplified expression is \( i \).

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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, denoted as \( i \), is a fundamental concept in the field of complex numbers. This unit is defined by the property \( i^2 = -1 \). This definition allows us to extend the real number system to include numbers that cannot be represented on the traditional number line. When combined with real numbers, \( i \) forms the basis for complex numbers.

  • Understanding \( i \) is crucial because it enables us to solve equations like \( x^2 + 1 = 0 \), which has no solution in the real numbers.
  • In these equations, the solutions often involve the square root of a negative number, which is where the imaginary unit comes into play.
  • For example, the square root of \(-1\) is defined to be \( i \), thus \( \sqrt{-1} = i \).
This unique property of the imaginary unit forms the backbone for complex arithmetic, allowing for a wider set of solutions and operations beyond the real numbers.
Powers of i
Powers of \( i \) exhibit a repeating cycle, crucial for simplifying expressions involving high powers of \( i \). Understanding this cycle aids in reducing such expressions easily.

  • The repetition cycle is:
    • \( i^1 = i \)
    • \( i^2 = -1 \)
    • \( i^3 = -i \)
    • \( i^4 = 1 \)
  • After this, the powers repeat: \( i^5 = i \), \( i^6 = -1 \), and so on.
  • Knowing the cycle allows you to determine the expression \( i^n \) for any positive integer \( n \) by dividing \( n \) by 4 and finding the remainder.
For example, for \( i^{17} \), divide 17 by 4 which gives a remainder of 1. Thus, \( i^{17} \equiv i^1 = i \). This process greatly simplifies computations and helps in comprehending complex mathematical concepts involving powers of \( i \).
Complex Number Simplification
Simplifying complex numbers involves reducing expressions with complex components into a single, more compact form. This is often necessary in mathematical calculations and engineering applications.

  • To simplify an expression like \( i^{17} \), use the powers of \( i \) cycle.
  • By determining the suitable power (from a cycle of 4 as seen above), the expression reduces to a simple form, either \( i \), \(-1\), \(-i\), or \(1\).
  • This process ensures that calculations remain manageable and easily interpretable.
The primary goal of complex number simplification is to leverage the consistent patterns, especially of the imaginary unit, to convert complex expressions into a format that is easier to work with. By understanding these patterns, you ensure clarity and efficiency in problem-solving, highlighting the elegance and utility of complex numbers in various fields.

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Most popular questions from this chapter

Solve each of the following equations for all complex solutions. $$ z^{7}=3 $$

Sketch a graph of the polar equation. $$ r=4 \sin (5 \theta) $$

Rewrite each complex number from polar form into \(a+b i\) form. $$ 4 e^{4 i} $$

Eliminate the parameter \(t\) to rewrite the parametric equation as a Cartesian equation. $$ \left\\{\begin{array}{l} x(t)=t^{5} \\ y(t)=t^{10} \end{array}\right. $$

Rewrite each complex number into polar \(r e^{i \theta}\) form. $$ -2+3 i $$
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