Problem 24 - Advanced Engineering Mathematics

Question

$$(2+3 i)(-i)^{2}=-2-3 i$$

Step-by-Step Solution

Answer

True, the equation holds as both sides equal \(-2 - 3i\).

Step 1: Understand the Problem

The given expression is \((2+3i)(-i)^2 = -2 - 3i\). We need to simplify the left-hand side and verify if it equals the right-hand side.

Step 2: Apply Exponentiation of Complex Number

Calculate \((-i)^2\). Since \(i\) is the imaginary unit where \(i^2 = -1\), we have \((-i)^2 = (-1)^2 \times i^2 = 1 \times (-1) = -1\).

Step 3: Simplify the Expression

Substitute the result from Step 2 into the expression: \((2+3i)(-1)\). This simplifies to \(-2 - 3i\) by distributing \(-1\) across the terms \(2+3i\).

Step 4: Verify the Solution

Compare the simplified left-hand side \(-2 - 3i\) with the given right-hand side \(-2 - 3i\). Since they match, the given equation holds true.

Key Concepts

Imaginary Unit

The concept of the imaginary unit is fundamental in understanding complex numbers. Represented as \(i\), the imaginary unit is defined with the key property \(i^2 = -1\). This definition may seem abstract, but it allows us to solve equations that do not have real solutions. For example, the equation \(x^2 + 1 = 0\) has no real solution, as no real number squared will equal \(-1\). However, with the imaginary unit, \(x = i\) becomes a valid solution.

• \(i\) is the foundation of imaginary numbers.
• Complex numbers are expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.

Understanding \(i\) is crucial because it extends the real number system into the realm of complex numbers, giving us powerful tools to tackle a wider range of mathematical problems.

Exponentiation

Exponentiation in complex numbers involves raising a complex number to a power, utilizing the rules of exponentiation known for real numbers. In this instance, we focus on exponentiating the imaginary unit. In our original exercise, we have \((-i)^2\). To solve this, we must understand that \(i^2 = -1\). To further break it down:- \((-i)^2\) can be rewritten using multiplication as \((-1)^2 \times i^2\).- Since \((-1)^2 = 1\) and \(i^2 = -1\), this results in \(1 \times (-1) = -1\).This step shows how exponentiation works the same way as it does for real numbers, with the added layer of understanding that \(i^2\), or the square of the imaginary unit, always equals \(-1\). This knowledge is vital for simplifying complex expressions and verifying complex number equations.

Simplification

Simplification involves breaking down complex expressions into more manageable forms. In the given exercise, after finding that \((-i)^2 = -1\), we need to simplify \((2+3i)(-1)\). This process is straightforward:- Distribute the multiplication with \(-1\) across each term in the complex number \(2 + 3i\).- This becomes \(-1 \times 2 + (-1) \times 3i = -2 - 3i\).By following the distributions:- The real part of the number, \(2\), becomes \(-2\).- The imaginary part, \(3i\), turns into \(-3i\).Simplifying such expressions allows us to see the effect of operations directly on complex numbers, making it easier to analyze and verify our equations.

Verification of Equations

Verification of equations is a critical step in ensuring that our simplifications and calculations are correct. For the equation \((2+3i)(-i)^2 = -2 - 3i\), we begin by simplifying the left-hand side to \(-2 - 3i\). Finally, comparing this simplified form with the right-hand side, \(-2 - 3i\), shows that both are identical. This confirms that our operations were conducted correctly.

• Ensure each mathematical step follows from the last.
• Cross-check the result with the original equation's right-hand side.

The process of verifying helps in reinforcing the accuracy of solutions and understanding of the principles involved with complex numbers.