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Chapter 9: Problem 29

Find each product. $$ (3+6 i)(3-6 i) $$

Short Answer

Expert verified
45

Step by step solution

01

- Identify the form

Recognize that the given expression \((3+6i)(3-6i)\) is in the form of a product of two complex conjugates.
02

- Apply the formula for complex conjugates

Recall that the product of two complex conjugates \((a+bi)(a-bi)\) is \(a^2 + b^2\). Here, \(a=3\) and \(b=6\).
03

- Square the real and imaginary parts

First, square the real part: \(3^2 = 9\). Next, square the imaginary part: \(6^2 = 36\).
04

- Add the squared parts together

Combine the results from Step 3: \(9 + 36 = 45\).

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Key Concepts

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

Complex Conjugates
When working with complex numbers, a complex conjugate plays an important role. A complex conjugate of a complex number is created by changing the sign of the imaginary part.
For example, if you have a complex number in the form of \(a+bi\), its complex conjugate will be \((a-bi)\).
In other words, they look almost the same but have their imaginary parts (the \(bi\) part) with opposite signs.
Complex conjugates are particularly useful in multiplication and division of complex numbers.
In the exercise, \(3+6i\) and \(3-6i\) are complex conjugates.
Product of Complex Numbers
Multiplying complex numbers can seem tricky, but there are specific rules that make the process easier.
When multiplying two complex conjugates, like \(a+bi\) and \((a-bi)\), a special formula applies: \((a+bi)(a-bi) = a^2 + b^2\).
Notice that when you multiply out the terms, the imaginary parts actually cancel each other out.
This leaves you with the simpler result of the squares of the real and imaginary parts added together.
Applying this to our given problem, \( (3+6i)(3-6i) \), we see that it matches the form of the formula, making our computation straightforward.
Squaring Real and Imaginary Parts
To find the product of the complex conjugates, we need to compute the squares of the real and imaginary parts.
Here's the step-by-step:
\(a=3\) is the real part and \(b=6\) is the imaginary part.
We square these parts individually: Real part squared: \3^2 = 9\ and Imaginary part squared: \6^2 = 36\.
Finally, add these results together:
\[9 + 36 = 45\]
Therefore, \( (3+6i)(3-6i) \) equals 45. This final sum is always a non-negative real number when dealing with products of complex conjugates.

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Most popular questions from this chapter

Solve each problem. Find two numbers whose sum is 80 and whose product is a maximum. (Hint: Let \(x\) represent one of the numbers. Then \(80-x\) represents the other. A quadratic function represents their product.)

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 3 x^{2}-2 x+3=0 $$

Concept Check Which step is an appropriate way to begin solving the quadratic cquation \(2 x^{2}-4 x=9\) by completing the square? A. Add 4 to each side of the equation. B. Factor the left side as \(2 x(x-2)\) C. Factor the left side as \(x(2 x-4)\) D. Divide each side by 2

Write each quotient in standard form. $$ \frac{13-14 i}{12+24 i} $$

Write each quotient in standard form. $$ \frac{-i}{1+2 i} $$
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