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Chapter 1: Problem 92

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

Short Answer

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

Step by step solution

01

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.
02

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: \(-3i \cdot 3i = -9i^2\)
03

Simplify \(i^2\) Term

Recall that \(i^2 = -1\). Thus, \(-9i^2 = -9(-1) = 9\).
04

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: \(-12i - 12i = -24i\)
05

Write in Standard Form

The expression simplifies to \(-7 - 24i\), which is in the standard form \(a + bi\).
06

Final Answer Confirmation

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

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Key Concepts

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

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鈥檚 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鈥檛 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.

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Most popular questions from this chapter

Both the La Plata river dolphin (Pontoporia blainvillei) and the sperm whale ( Physeter macrocephalus) belong to the suborder Odontoceti (individuals that have teeth). A La Plata river dolphin weighs between 30 and \(50 \mathrm{~kg}\), whereas a sperm whale weighs between 35,000 and \(40,000 \mathrm{~kg} .\) A sperm whale is \(\quad\) order(s) of magnitude heavier than a La Plata river dolphin.

This problem relates to self-thinning of plants (Example 8). If we plot the logarithm of the over age weight of a plant, \(\log w\), against the logarithm of the density of plants, \(\log d\) (base 10 ), a straight line with slope \(-3 / 2\) results. Find the equation that relates \(w\) and \(d\), assuming that \(w=1 \mathrm{~g}\) when \(d=10^{3} \mathrm{~m}^{-2}\).

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=9 x^{-3} $$

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Explain how the following functions can be obtained from \(y=e^{x}\) by basic transformations: (a) \(y=e^{-x}-1\) (b) \(y=-e^{x}+1\) (c) \(y=-e^{x-3}-2\)
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