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Chapter 3: Problem 44

Multiply. Write the answer in standard form. \((8+3 i)^2\)

Short Answer

Expert verified
The squared result of the complex number (8 + 3i), in standard form, is \(55 + 48i\).

Step by step solution

01

Write Out the Expression

First, we rewrite the equation \((8 + 3i)^2\) as \((8 + 3i) * (8 + 3i)\).
02

Apply FOIL Method

Now, using the FOIL method, we get: First terms: \(8 * 8 = 64\),Outer terms: \(8 * 3i = 24i\),Inner terms: \(3i * 8 = 24i\),Last terms: \(3i * 3i = 9i^2\). Thus we get the equation: \(64 + 24i + 24i + 9i^2\).
03

Simplify the i terms

Now, we simplify the equation. In the realm of complex numbers, \(i^2 = -1\). So, we substitute \(i^2\) with \(-1\) in our equation to get: \(64 + 24i + 24i - 9\). Combining like terms gives us: \(55 + 48i\).

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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 an acronym that stands for First, Outer, Inner, Last. It's a technique used to multiply two binomials鈥攁 type of algebraic expression containing two terms. Let's look at how it works using our exercise as an example. When we have , \((8+3i)^2\), we rewrite it as , \((8+3i)(8+3i)\).
To apply the FOIL method, we multiply the First terms (8 and 8), the Outer terms (8 and 3i), the Inner terms (3i and 8), and finally the Last terms (3i and 3i). This method ensures that all parts of the binomials are multiplied together. Mastering the FOIL method is crucial for simplifying complex expressions, particularly when involving imaginary numbers.
Standard Form
Standard form for a complex number is written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) being the square root of -1. In the context of our exercise, after multiplying and combining like terms using the FOIL method, we aim to express the result in this format. With the final simplified expression being \( 55 + 48i \), we have reached the standard form. This expression clearly shows the real part, 55, and the imaginary part, \( 48i \). Keeping complex numbers in standard form allows for easy addition, subtraction, and comparison with other complex numbers.
Simplify Complex Expressions
Simplifying complex expressions becomes straightforward when you understand the rules for imaginary numbers. In our exercise, after applying the FOIL method, we obtain terms containing \( i \), the imaginary unit. One such term is \( 9i^2 \). Since \( i^2 = -1 \), we replace \( 9i^2 \) with -9. This step is essential for simplifying complex expressions since it involves reducing the expression to its most basic form. After substitution, we combine like terms, which means adding or subtracting real numbers with real numbers, and imaginary terms with imaginary terms. That's how we simplify the initial expression to get the easy-to-understand standard form.
Imaginary Numbers
Imaginary numbers are a fascinating and fundamental concept in complex number algebra. They are defined as multiples of \( i \), which is the square root of -1鈥攁 number that cannot exist within the set of real numbers. In our example, \( i \) becomes crucial when we square \( 3i \) to obtain \( 9i^2 \). Here, we apply the definition that \( i^2 = -1 \). This characteristic allows us to simplify expressions that would otherwise not have real solutions. Understanding imaginary numbers and their properties is pivotal when solving quadratic equations and performing operations with complex numbers.

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Most popular questions from this chapter

In Exercises 25-36, solve the equation by completing the square. \(x^2+6 x+3=0\)

Justify each step in performing the operation. $$ \begin{aligned} (3&+2 i)(7-4 i) \\ &=21-12 i+14 i-8 i^2 \\ &=21+2 i-8(-1) \\ &=21+2 i+8 \\ &=29+2 i \end{aligned} $$

Solve the inequality. Graph the solution. \(\frac{n}{3}+6>1\)

Find the minimum value or maximum value of the function. Then describe where the function is increasing and decreasing. (Section 2.2) \(f(x)=-(x-3)(x+7)\)

Solve the inequality. Graph the solution. \(2 x-3<5\)
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