Problem 38 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified result is 25.

Step 1: 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.

Step 2: 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.

Step 3: Compute the Squares

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

Step 4: Add the Squares

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

Step 5: Calculate the Final Result

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

Key Concepts

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.