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Chapter 11: Problem 3

Simplify (a) \(-j^{2},(b)(-j)^{2},(c)(-j)^{3}\), (d) \(-j^{3}\).

Short Answer

Expert verified
Question: Simplify the following complex number expressions: (a) -j^2, (b) (-j)^2, (c) (-j)^3, and (d) -j^3. Answer: (a) -j^2 = 1, (b) (-j)^2 = 1, (c) (-j)^3 = -j, (d) -j^3 = j.

Step by step solution

01

Understand the problem

We are given four expressions (a) -j^2, (b) (-j)^2, (c) (-j)^3, and (d) -j^3. Our goal is to simplify each expression using the property of the imaginary unit j^2 = -1 and exponent rules.
02

Simplify (a) -j^2

Using the property j^2 = -1, we can directly substitute -1 into the expression: -j^2 = -(-1) = 1.
03

Simplify (b) (-j)^2

In this expression, the entire complex number -j is squared, so we need to square both the real part and the imaginary part: (-j)^2 = (-1) * (j^2) = (-1) * (-1) = 1.
04

Simplify (c) (-j)^3

Here, we need to cube the entire complex number -j. We can use the property j^2 = -1 to help simplify: (-j)^3 = (-j) * (-j) * (-j) = (-1) * (-1) * j^3. To simplify j^3, we use the property j^2 = -1: j^3 = j * j^2 = j * (-1) = -j. So, (-j)^3 = (-1) * (-1) * (-j) = -j.
05

Simplify (d) -j^3

For this expression, only the imaginary part j is cubed, and then multiplied by the negative sign: -j^3 = -1 * (j^3). As in the previous step, we can simplify j^3: -j^3 = -1 * (-j) = j. The simplified results are: (a) -j^2 = 1 (b) (-j)^2 = 1 (c) (-j)^3 = -j (d) -j^3 = j

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Most popular questions from this chapter

Consider the complex number \(z=\cos \theta+\mathrm{j} \sin \theta\) (a) Write down the exponential form of this number. (b) By raising the exponential form to the power \(n\), and using one of the laws of indices, deduce De Moivre's theorem.

Solve each of the following equations leaving (a) \(z^{3}=-1\) (b) \(z^{3}=1\) (c) \(z^{3}-6 \mathrm{j}=0\) your answers in polar form: (a) \(z^{2}=1+\mathrm{j}\) (b) \(z^{2}=1-\mathrm{j}\) (c) \(z^{3}=-2+3 \mathrm{j}\)

A capacitor and resistor are placed in parallel. Show that the complex impedance of this combination is given by $$ \frac{1}{Z}=\frac{1}{R}+j \omega C $$ Find an expression for \(Z\).

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