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Chapter 3: Problem 64

Multiply as indicated. Write each product in standand form. $$(6-4 i)(6+4 i)$$

Short Answer

Expert verified
The product in standard form is 52.

Step by step solution

01

Recall the formula for the difference of squares

The expression \((a-b)(a+b)\) is a difference of squares and can be expanded as \(a^2 - b^2\). Note that \(i^2 = -1\).
02

Identify variables

In the given expression \((6-4i)(6+4i)\), identify \(a = 6\) and \(b = 4i\).
03

Apply the difference of squares formula

Using the formula from Step 1, compute \(a^2 - b^2\) where \(a = 6\) and \(b = 4i\). Substitute the values: \[a^2 = 6^2 = 36\,b^2 = (4i)^2 = 16i^2 = 16(-1) = -16\,a^2 - b^2 = 36 - (-16) = 36 + 16 = 52\].
04

Write the result in standard form

The product of \((6-4i)(6+4i)\) using the difference of squares is \(52\). This is already in standard form as there is no imaginary part.

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Key Concepts

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

Difference of Squares
The difference of squares is a mathematical expression used for factoring specific types of expressions. It can be recognized in the pattern
  • \((a-b)(a+b)\)
This pattern is known as the difference of squares because it can be expanded as
  • \(a^2 - b^2\)
The reason behind this name is that you're subtracting the square of one term from another. Let鈥檚 break it down: if you have two terms, \(a\) and \(b\), multiplying their sum and their difference results in the difference between their squares. This concept is not just theoretical; it has applications in simplifying expressions, especially when dealing with polynomials and complex numbers.
Imaginary Unit
When dealing with complex numbers, the imaginary unit, represented by \(i\), plays a crucial role. The imaginary unit is defined by its unique property:
  • \(i^2 = -1\)
This might seem strange at first because squaring a number usually yields a positive result. However, \(i\) is specifically designed to make sense of square roots of negative numbers, which don鈥檛 exist in the set of real numbers.
In complex numbers, the value of \(b\) in a difference of squares can be imaginary, such as \(4i\) in our original problem. This means when squared, it gives a negative result thanks to the property of \(i\):
  • \((4i)^2 = 16i^2 = 16(-1) = -16\)
Understanding \(i\) helps in managing calculations involving complex numbers.
Standard Form
The standard form for complex numbers is expressed as
  • \(a + bi\)
with \(a\) representing the real part, and \(b\) representing the imaginary part. When a complex number is written in standard form, it clearly shows the balance between the real and imaginary parts.
However, in the solution of the expression
  • \((6-4i)(6+4i)\)
we derived a result
  • \(52\)
which is purely a real number. This is already in standard form because the imaginary part is zero. When simplifying or solving expressions involving complex numbers, reaching or maintaining the standard form is often a key goal, ensuring interpretations are consistent and clear.

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Most popular questions from this chapter

For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverses $$3 i,-3 i$$

Solve each problem. The table shows a person's heart rate during the first 4 minutes after exercise has stopped. $$\begin{array}{|l|c|c|c|} \hline \text { Time (min) } & 0 & 2 & 4 \\ \hline \text { Heart rate (bpm) } & 154 & 106 & 90 \\ \hline \end{array}$$ (a) Find a formula \(f(x)=a(x-h)^{2}+k\) that models the data, where \(x\) represents time and \(0 \leq x \leq 4 .\) Use \((4,90)\) as the vertex. (b) Evaluate \(f(1)\) and interpret the result. (c) Estimate the times when the heart rate was from 115 to 125 beats per minute.

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Solve each equation for \(x\) and then for \(y .\) $$x^{2}+x y+y^{2}=0 \quad(x>0, y>0)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-12 x=-11$$
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