Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (3+4 i)(3-4 i) $$

Step-by-Step Solution

Answer

The result is \(25 + 0i\).

Step 1: Identify the operation

The problem requires us to multiply two complex numbers: \((3+4i)\) and \((3-4i)\). These are conjugates of each other.

Step 2: Apply the formula for multiplying conjugates

The multiplication of two conjugates \((a+bi)(a-bi)\) can be simplified using the formula: \((a^2 + b^2)\). In this case, \(a=3\) and \(b=4\).

Step 3: Substitute values and calculate

Using the formula \((a^2 + b^2)\), substitute the values: \[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \]

Step 4: Write the result as a complex number

The result \(25\) is real, so it can be expressed as a complex number without the imaginary part: \(25 + 0i\).

Key Concepts

Conjugates

When dealing with complex numbers, the concept of conjugates is quite important. A complex number is of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\), the imaginary unit satisfying \(i^2 = -1\).
The conjugate of a complex number \(a + bi\) is \(a - bi\). This means you simply change the sign of the imaginary part.

Conjugates are useful because they allow us to simplify complex expressions, especially when multiplying complex numbers.
Multiplying conjugates often results in a real number. This happens because when you multiply \((a + bi)\) by \((a - bi)\), the imaginary parts cancel out.

• Real parts add up: \(a \times a = a^2\)
• Imaginary parts cancel: \((bi)\times (-bi) = b^2i^2 = -b^2(-1) = b^2\)

This results in \(a^2 + b^2\), a simplified real number.

Multiplying Complex Numbers

Multiplying complex numbers can seem tricky at first, but it follows some straightforward rules. Consider two complex numbers, \((a+bi)\) and \((c+di)\).
To multiply them together, you distribute each part, much like in algebra with binomials:

• Multiply the real parts: \(a \times c\)
• Multiply the real part of the first and imaginary part of the second: \(a \times di\)
• Multiply the imaginary part of the first and real part of the second: \(bi \times c\)
• Multiply the imaginary parts: \(bi \times di = bdi^2 = -bd\)

Simplifying all these terms, you combine like terms to get a new complex number: \((ac - bd) + (ad + bc)i\).
It's crucial to remember that whenever you multiply parts that include \(i\), keep in mind \(i^2 = -1\) which helps convert a term back into a real number.

Simplifying Complex Expressions

Simplifying complex expressions is all about reducing them to their simplest form while preserving their mathematical meaning. When you multiply conjugates like in this exercise, the result simplifies due to the cancellation of imaginary parts.
Let's see how this works using the expression \((3+4i)(3-4i)\).
Recognize these as conjugates and apply the formula:

• Multiply and simplify: \(a^2 + b^2\) using \(a = 3\) and \(b = 4\).
• This gives \(3^2 + 4^2 = 9 + 16 = 25\).

This process shows how multiplying conjugates simplifies a complex pair into a real number, in this case, 25.
By understanding these steps, you can handle similar operations, simplifying complex numbers efficiently without much hassle.