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Chapter 5: Problem 38

Simplify. $$ (3+4 i)(3-4 i) $$

Short Answer

Expert verified
The simplified result is 25.

Step by step solution

01

Identify the Expressions

The original expression is \((3 + 4i)(3 - 4i)\). This expression represents a complex number multiplication where \(a + bi\) and \(a - bi\) are conjugates.
02

Apply the Formula for Conjugates

The product of a complex number and its conjugate is given by the formula \((a + bi)(a - bi) = a^2 + b^2\). Identify \(a = 3\) and \(b = 4\) for this particular exercise.
03

Compute the Squares

Calculate \(a^2\) and \(b^2\), where \(a = 3\) and \(b = 4\). Thus, \(a^2 = 3^2 = 9\) and \(b^2 = 4^2 = 16\).
04

Add the Squares

Use the results of \(a^2\) and \(b^2\) to find the expression's value: \(a^2 + b^2 = 9 + 16\).
05

Calculate the Final Result

Add the squares: \(9 + 16 = 25\). This is the simplified result of \((3 + 4i)(3 - 4i)\).

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Key Concepts

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

Understanding the Conjugate
In complex numbers, the conjugate of a complex number changes the sign of the imaginary part. Think of a complex number as taking the form \(a + bi\). Its conjugate would then be \(a - bi\). This simple change affects how we can use conjugates in multiplication.
  • Opposite Signs: The imaginary components have opposite signs.
  • Real Part Remains: The real part \(a\) remains unchanged in the conjugate.
  • Impact on Multiplication: Multiplying a complex number by its conjugate simplifies calculations and can help eliminate the imaginary part.
Conjugates are particularly useful because multiplying a complex number by its conjugate results in a real number. This phenomenon is key in the simplification process.
Complex Number Multiplication
Complex number multiplication involves applying the distributive property, similar to multiplying binomials in algebra. When multiplying \((3 + 4i)\) by \((3 - 4i)\), you distribute each part of the first binomial by each part of the second binomial. Here is what we need to focus on:
  • Distributing Terms: Multiply each part separately: \(3\times3\), \(3\times(-4i)\), \(4i\times3\), and \(4i \times (-4i)\).
  • Combining Like Terms: Sum the results to combine real and imaginary parts separately.

Using conjugates offers a shortcut because certain terms cancel out, leaving you with the simple formula \(a^2 + b^2\), where \(a\) and \(b\) are the parts of the complex number.
Simplification with Conjugates
The simplification process becomes notably easier when dealing with conjugates. The unique property of a complex number conjugate, where the imaginary parts cancel, simplifies to just adding squares. Given \((3+4i)(3-4i)\), we already determined that:
  • Real Part: Identify \(a = 3\) and \(b = 4\).
  • Squares Calculation: Compute \(a^2 = 9\) and \(b^2 = 16\).
  • Add Squares: Combine these to get \(a^2 + b^2 = 25\).

Therefore, the simplification boils down to just the addition of these squares, leading to a real number, 25. This approach provides an efficient way to handle the multiplication of complex numbers when using conjugates.

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Most popular questions from this chapter

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. What is the maximum profit the team can make with this fund-raiser, and how many passengers will it take to achieve this maximum?

Solve each equation by using the Square Root Property. \(x^{2}+18 x+81=25\)

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=\frac{1}{2} x^{2}+6 x+18 $$

Solve each equation by using the method of your choice. Find exact solutions. \(x^{2}+9=8 x\)

Find the exact solutions of \(2 i x^{2}-3 i x-5 i=0\) by using the Quadratic Formula.
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