Problem 92 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((-4-3 i)(4+3 i)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-7 - 24i\), which is the standard form.

Step 1: Identify the Expression Form

The expression \((-4-3i)(4+3i)\) is in the form of \((a+bi)(c+di)\). Our goal is to simplify this product.

Step 2: Apply the FOIL Method

Use the FOIL method (First, Outer, Inner, Last) to expand the product:- First: \(-4 \cdot 4 = -16\)- Outer: \(-4 \cdot 3i = -12i\)- Inner: \(-3i \cdot 4 = -12i\)- Last: \(-3i \cdot 3i = -9i^2\)

Step 3: Simplify \(i^2\) Term

Recall that \(i^2 = -1\). Thus, \(-9i^2 = -9(-1) = 9\).

Step 4: Combine Like Terms

Add together the expanded components from Step 2:\(-16 - 12i - 12i + 9\).Combine like terms:- Real part: \(-16 + 9 = -7\)- Imaginary part: \(-12i - 12i = -24i\)

Step 5: Write in Standard Form

The expression simplifies to \(-7 - 24i\), which is in the standard form \(a + bi\).

Step 6: Final Answer Confirmation

Review each step to ensure calculations are correct, and confirm the final answer is in the form \(a+bi\).

Key Concepts

FOIL Method

The FOIL method is a powerful tool for expanding the product of two binomials. This is particularly useful when dealing with complex numbers in the form \((a+bi)(c+di)\). FOIL stands for First, Outer, Inner, Last. Here’s how each step works with our original exercise example:

• First: Multiply the first terms in each binomial: \(-4 \cdot 4 = -16\).
• Outer: Multiply the outer most terms in the expression: \(-4 \cdot 3i = -12i\).
• Inner: Multiply the inner most terms: \(-3i \cdot 4 = -12i\).
• Last: Multiply the last terms in the binomials: \(-3i \cdot 3i = -9i^2\).

By following these steps, you properly organize and simplify the expression. Don’t forget, when you reach the last step, \(-9i^2\) turns into 9because \(i^2 = -1\). This is crucial for correctly expanding your expression.

Imaginary Unit

The imaginary unit, denoted as \(i\), plays a critical role in complex numbers. It is defined by the property \(i^2 = -1\). Understanding this concept is key for simplifying expressions involving complex numbers.

In our exercise, the term \(-9i^2\) appears. By substituting \(i^2\) with \(-1\), the term simplifies to \(9\). This substitution transforms complex products into real numbers, highlighting why understanding \(i\) is essential.

Always remember:

• When you see \(i^2\), replace it with \(-1\).
• Pay attention to how imaginary terms interact with real terms in a product.
• Recognize \(i\) as the building block for creating and simplifying complex scenarios.

Once you're comfortable with \(i\), handling complex expressions becomes much simpler.

Standard Form

When working with complex numbers, it's important to express them in the standard form, \(a + bi\). This format consists of two parts:

• The real part, \(a\), a regular real number, easy to identify and isolated from any imaginary component.
• The imaginary part, \(bi\), which includes the imaginary unit \(i\).

After using the FOIL method on our exercise, our expression becomes \(-7 - 24i\). Here, \(-7\) is the real part, and \(-24i\) is the imaginary part. To express the result in standard form, simply combine these terms clearly, as we've done.

Why is standard form helpful?

• It provides a clear and organized way to present complex numbers.
• Makes addition, subtraction, and comparison of complex numbers simpler.
• Helps in easily identifying the magnitude and direction if represented on a complex plane.

Develop a habit of converting complex expressions to this form for clarity and ease of further calculations.