Questions: Perform the indicated operation. (5+9 i)(5-9 i) (5+9 i)(5-9 i)= (Simplify your answer. Type your answer in the form a + bi.)

Perform the indicated operation. (5+9 i)(5-9 i) (5+9 i)(5-9 i)= (Simplify your answer. Type your answer in the form a + bi.)
Transcript text: / College Algebra with Support / FA24 Complex Numbers Questi Perform the indicated operation. \[ \begin{array}{l} (5+9 i)(5-9 i) \\ (5+9 i)(5-9 i)= \end{array} \] $\square$ (Simplify your answer. Type your answer in the form a + bi.)
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Solution

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Solution Steps

To solve the problem of multiplying two complex numbers, we can use the formula for the product of conjugates. The product of a complex number and its conjugate is a real number, calculated as \(a^2 + b^2\), where \(a\) and \(b\) are the real and imaginary parts, respectively. In this case, the numbers are \(5 + 9i\) and \(5 - 9i\).

Step 1: Identify the Complex Numbers

We are given the complex numbers \( (5 + 9i) \) and \( (5 - 9i) \).

Step 2: Apply the Product of Conjugates Formula

The product of a complex number and its conjugate can be calculated using the formula: \[ (5 + 9i)(5 - 9i) = 5^2 + 9^2 \]

Step 3: Calculate the Squares

Calculating the squares: \[ 5^2 = 25 \quad \text{and} \quad 9^2 = 81 \]

Step 4: Sum the Results

Now, we sum the results: \[ 25 + 81 = 106 \]

Final Answer

Thus, the simplified result of the operation is: \[ \boxed{106} \]

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