Question

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

Short answer

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

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.

Simplify (a) -j^2

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

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.

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.

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

Key concepts

Imaginary Unit

The imaginary unit is a special concept in mathematics, primarily used in the study of complex numbers. It is denoted by the symbol \(j\) (or sometimes \(i\) depending on the engineering or mathematics context). The defining property of the imaginary unit is that its square equals

• \(j^2 = -1\)

This property is fundamental because it allows us to represent numbers that aren't real—hence the name "imaginary." Essentially, imaginary numbers are multiples of this imaginary unit. They help us in various fields like electrical engineering, solving quadratic equations with negative discriminants, and even advanced physics.
Understanding the nature of \(j\) allows us to perform arithmetic with complex numbers and is central to this exercise.

Exponent Rules

When dealing with the powers of imaginary numbers, exponent rules play a vital role. Knowing how to manage exponents with complex numbers allows for efficient simplification.

• When raising a number to a power, distribute the power to both parts of a complex term.
• Apply the property \(j^2 = -1\) wherever possible.

For example, the expression \((-j)^2\) can be resolved by recognizing it as the product \(( -1 ) \times ( j^2 )\). Each part is raised to the power, transforming \(j^2\) into \(-1\), leading directly to the solution.
Remember these rules ease the simplification process, letting us break down potentially complex problems into more manageable pieces. Mastering exponent rules is key to handling equations involving complex numbers and imaginary units efficiently.

Simplification Steps

Simplifying expressions involving complex numbers requires a strategic approach. Each problem starts with identifying the power component involved and any related properties. Here are some simplification steps that can be useful:

• Recognize the imaginary unit⟐ Know that \( j^2 = -1 \) so you can immediately replace \( j^2 \) by \(-1\) in your expressions.
• Simplify powers individually⟐ Break complex expressions into smaller parts when possible. For example, for \((-j)^3\), consider the nested powers as \((-1) \times (-1) \times j^3\).

These steps help reduce complex expressions to more straightforward numbers. They make it easier to understand and handle expressions such as \(-j^3\), which can initially seem daunting but break down into simpler operations using these methods. Always aim for step-by-step reductions, ensuring clarity and accuracy at each stage.