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Chapter 14: Problem 1

Simplify: (a) \(j^{3}\) (b) \(j^{5}\) (c) \(j^{12}\) (d) \(j^{14}\).

Short Answer

Expert verified
(a) \( -j \), (b) \( j \), (c) \( 1 \), (d) \( -1 \).

Step by step solution

01

Understanding the imaginary unit

The imaginary unit is denoted as \( j \), and it is defined as \( j = \sqrt{-1} \). Essential powers of \( j \) are: \( j^2 = -1 \), \( j^3 = -j \), and \( j^4 = 1 \). These powers repeat every four terms, forming a cycle.
02

Simplifying \( j^3 \)

From the cycle of powers, we know that \( j^3 = -j \). This is an immediate result from the given recurring cycle of powers of \( j \).
03

Simplifying \( j^5 \)

Use the cycle to simplify \( j^5 \). \( j^5 = j^{4+1} = (j^4)\cdot(j^1) = 1 \cdot j = j \).
04

Simplifying \( j^{12} \)

Since \( j^4 = 1 \), we can express \( j^{12} \) as \( (j^4)^3 = 1^3 = 1 \).
05

Simplifying \( j^{14} \)

Decompose \( j^{14} \) into \( j^{12+2} = (j^{12})(j^2) = 1 \cdot (-1) = -1 \).

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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 is a fundamental concept in mathematics that helps extend the real number system to include roots of negative numbers. It is represented by the symbol \( j \), especially in engineering fields, where \( i \) is commonly used in pure mathematics. The imaginary unit is defined by the equation \( j = \sqrt{-1} \). This definition is crucial because it allows us to work with numbers that are not on the traditional real number line.

Here are some important points about the imaginary unit:
  • It provides a way to express solutions to equations like \( x^2 = -1 \).
  • It forms the basis of complex numbers, which are expressed as \( a + bj \), where \( a \) and \( b \) are real numbers.
  • It is used extensively in various fields, from electrical engineering to control systems.
Understanding the imaginary unit is essential for grasping more advanced concepts in mathematics and its applications.
Powers of Imaginary Numbers
The powers of the imaginary unit \( j \) follow a specific repeating pattern after every four terms. Knowing this cycle can simplify computations involving complex numbers.

The cycle is as follows:
  • \( j^1 = j \)
  • \( j^2 = -1 \)
  • \( j^3 = -j \)
  • \( j^4 = 1 \)
After \( j^4 \), the powers repeat, so \( j^5 = j \), \( j^6 = -1 \), and so on. This cyclical pattern is beneficial when simplifying powers of \( j \).

For example, to simplify \( j^{14} \), you can express it as \( j^{12+2} = (j^4)^3 \, j^2 \), knowing that \( j^4 = 1 \) and \( j^2 = -1 \), which results in \( -1 \). Understanding how powers of \( j \) cycle can save time and simplify calculations.
Simplification of Expressions
Simplifying expressions involving the imaginary unit involves using the properties and cycle of powers of \( j \). When faced with an expression like \( j^{14} \), the strategy revolves around breaking it down using the cyclic nature of powers of \( j \).

Here's how you can simplify some expressions efficiently:
  • Recognize the power pattern. Knowing \( j^4 = 1 \) helps break down larger powers of \( j \) into simpler parts.
  • Decompose the expression into components using this pattern. For example, \( j^{14} \) can be decomposed into \( j^{12} \times j^2 \), simplifying directly using known values.
  • Recombine and simplify the result using the cycle: \( j^{12} = 1 \) and \( j^2 = -1 \) gives \( j^{14} = -1 \).
By following these steps, you can simplify complex expressions involving the imaginary unit effectively, making conversion of complicated powers into manageable expressions possible.

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Most popular questions from this chapter

Find the modulus of \(z=(2-j)(5+j 12) /(1+j 2)^{3}\).

Given that \(z_{1}=R_{1}+R+j \omega L ; z_{2}=R_{2} ; z_{3}=\frac{1}{j \omega C_{3}} ;\) and \(z_{4}=R_{4}+\frac{1}{j \omega C_{4}} ;\) and also that \(z_{1} z_{3}=z_{2} z_{4}\), express \(R\) and \(L\) in terms of the real constants \(R_{1}, R_{2}\) \(R_{4}, C_{3}\) and \(C_{4}\)

]If \(z=\frac{a+j b}{c+j d}\), where \(a, b, c\) and \(d\) are real quantities, show that (a) if \(z\) is real then \(\frac{a}{b}=\frac{c}{d}\) and (b) if \(z\) is entirely imaginary then \(\frac{a}{b}=-\frac{d}{c}\). ]

If \(z_{1}=2+j, z_{2}=-2+j 4\) and \(\frac{1}{z_{3}}=\frac{1}{z_{1}}+\frac{1}{z_{2}}\), evaluate \(z_{3}\) in the form \(a+j b\) If \(z_{1}, z_{2}, z_{3}\) are represented on an Argand diagram by the points \(P, Q, R\), respectively, prove that \(\mathrm{R}\) is the foot of the perpendicular from the origin on to the line PQ.

If \(\frac{R_{1}+j \omega L}{R_{3}}=\frac{R_{2}}{R_{4}-j \frac{1}{\omega C}}\), where \(R_{1}, R_{2}, R_{3}, R_{4}, \omega, L\) and \(C\) are real, show that \(L=\frac{C R_{2} R_{3}}{\omega^{2} C^{2} R_{4}^{2}+1}\)
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