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Chapter 6: Problem 3

Calculate the given expression. \((2+3 i) \cdot(-3-2 i)\)

Short Answer

Expert verified
The result is \(-13i\).

Step by step solution

01

- Identify the given complex numbers

The given complex numbers are \(2 + 3i\) and \(-3 - 2i\.\)
02

- Apply the distributive property (FOIL method)

Expand the expression \((2 + 3i) \cdot (-3 - 2i)\) by multiplying each term: \((2) \cdot (-3) + (2) \cdot (-2i) + (3i) \cdot (-3) + (3i) \cdot (-2i)\)
03

- Multiply the real and imaginary parts

Calculate each multiplication separately:\(2 \cdot (-3) = -6\)\(2 \cdot (-2i) = -4i\)\(3i \cdot (-3) = -9i\)\(3i \cdot (-2i) = -6i^2\)
04

- Simplify the products

Combine the real and imaginary parts: \(-6 - 4i - 9i - 6i^2\)Recall that \(i^2 = -1\), so \(-6i^2 = -6(-1) = 6\)
05

- Combine like terms

Sum the real parts and the imaginary parts separately: Real: \(-6 + 6 = 0\)Imaginary: \(-4i - 9i = -13i\)So, the expression simplifies to \(0 - 13i = -13i\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The FOIL Method
When multiplying two complex numbers, we use the FOIL method. FOIL stands for First, Outer, Inner, Last. This method helps us distribute each term step-by-step.

Follow these steps for the expression \( (2 + 3i) \cdot\ (-3 - 2i) \):

1. **First:** Multiply the first terms: 2 \cdot\ -3 = -6.
2. **Outer:** Multiply the outer terms: 2 \cdot\ -2i = -4i.
3. **Inner:** Multiply the inner terms: 3i \cdot\ -3 = -9i.
4. **Last:** Multiply the last terms: 3i \cdot\ -2i = -6i^2.

Put it together: \( -6 - 4i - 9i - 6i^2 \).

The FOIL method breaks the problem into simple steps, making it easier to handle the multiplication.
The Imaginary Unit
The imaginary unit is essential when dealing with complex numbers. It's represented by the letter i.

By definition, i is the square root of -1, so \( i^2 = -1 \).

Understanding \( i^2 \) is crucial because, in our multiplication, we encountered \( -6i^2 \.\)

Replace \( i^2 \) with -1 to get: -6(-1) = 6.

Knowing how to handle the imaginary unit enables us to simplify expressions accurately.
Real and Imaginary Parts
In a complex number, like 2 + 3i, we have two parts:

1. **Real part**: The real number without i, in this case, 2.
2. **Imaginary part**: The number with i, here, 3i.

During multiplication, keep real and imaginary parts separate. After expanding and simplifying, combine the real parts and the imaginary parts:

\( -6 + 6 - 4i - 9i \)

Real parts: \( -6 + 6 \) = 0.
Imaginary parts: \( -4i - 9i \) = -13i.

Finally, our simplified expression isi.e., -13i.

Understanding the distinction between real and imaginary parts makes solving such problems more systematic.

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Most popular questions from this chapter

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