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Chapter 12: Problem 25

Simplify each of the given expressions. (a) \(-j^{6}\) (b) \((-j)^{6}\)

Short Answer

Expert verified
(a) 1, (b) -1

Step by step solution

01

Simplify the first problem

The expression is \(-j^{6}\). Since \(j\) is the imaginary unit where \(j^2 = -1\), we can find powers of \(j\) by reducing them to one of four principal values: \(j^1 = j\), \(j^2 = -1\), \(j^3 = -j\), and \(j^4 = 1\).We calculate:\[j^6 = (j^4) imes (j^2) = 1 imes (-1) = -1\]So, \(-j^{6}\) is:\[-(-1) = 1\].
02

Simplify the second problem

The expression is \((-j)^{6}\). Here, we need to account for the negative sign being inside the parentheses. We need to expand using the fact that \((-j)^6 = (-1)^6 imes j^6\)Since \((-1)^6 = 1\) (as any negative number raised to an even power yields a positive result) and from Step 1 we have \(j^6 = -1\): \[(-j)^6 = 1 imes (-1) = -1\].

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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 an essential concept in complex numbers, represented by the symbol \( j \) (often denoted as \( i \) in mathematics, but \( j \) is used in engineering). This unit is pivotal because it allows us to work with numbers that have no real counterpart on the traditional number line.

  • The defining property of the imaginary unit is that \( j^2 = -1 \), which directly sets it apart from real numbers.
  • This property allows mathematical modeling of scenarios we couldn't address with just real numbers, such as electrical engineering problems and signal processing.
  • By understanding the imaginary unit, you can extend your arithmetic from the real line to the complex plane, opening up a broader realm of mathematical solutions.
Powers of i
When dealing with the imaginary unit \( j \), it's important to recognize the pattern of its powers to simplify expressions efficiently.

  • The powers of \( j \) cycle through four values: \( j^1 = j \), \( j^2 = -1 \), \( j^3 = -j \), and \( j^4 = 1 \).
  • After \( j^4 \), the pattern repeats itself, so \( j^5 = j \), \( j^6 = -1 \), and so on.
  • To simplify any power of \( j \), divide the exponent by 4 and use the remainder to find the equivalent among the four basic powers.
For example, in the expression \( j^6 \), you divide 6 by 4 to get a remainder of 2, indicating \( j^6 = j^2 = -1 \). Understanding this cyclical pattern is key to mastering complex numbers.
Simplifying Expressions
Simplifying expressions involving complex numbers often comes down to understanding how the imaginary unit behaves and how its powers interact.

  • Always pay attention to whether the imaginary unit is inside or outside parentheses, as it affects the simplification.
  • For \(-j^6\), notice that only the power of \( j \) affects the outcome, leading to calculation and simplification directly without additional algebra.
  • When parentheses are involved, like in \((-j)^6\), expand using properties of exponents: consider both the negative sign and the imaginary unit.
When dealing with expressions like \((-j)^6\), remember that any negative number raised to an even power will become positive, influencing the final sign of your result. This distinction ensures you handle each expression component accurately, leading to correct simplifications.

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Most popular questions from this chapter

Express the given complex numbers in polar and rectangular forms. $$1724 e^{2.391 j}$$

$$\text {solve the given problems.}$$ Multiply \(-3+j\) by its conjugate.

Represent each complex number graphically and give the rectangular form of each. $$2.50\left(\cos 315.0^{\circ}+j \sin 315.0^{\circ}\right)$$

answer or explain as indicated. Explain how to show that the reciprocal of the imaginary unit is the negative of the imaginary unit.

Express the given complex numbers in polar and rectangular forms. $$2.50 e^{3.84 j}$$
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