Problem 15 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (3+5 i)(3-5 i) $$

Step-by-Step Solution

Answer

Thus, the product of \(3+5i\) and \(3-5i\) in standard form is 34.

Step 1: Identify and multiply the real parts

First, identify the real parts of the complex number, which are the numbers without the imaginary unit 'i'. These are 3 in both cases. Thus, start by multiplying these together to get \(3 \times 3 = 9\).

Step 2: Multiply the imaginary parts

Next, identify and multiply the coefficients of the imaginary parts together. These are +5 and -5. Hence, the multiplication gives \((+5i) \times (-5i) = -25i^2\). Remember that \(i^2\) is -1, therefore \(-25i^2\) simplifies to 25.

Step 3: Combine the real and imaginary parts

Finally, you combine the results of the real and the imaginary parts. The real part from the multiplication is 9 and the real part from the multiplication of the imaginary parts is 25. The result is \(9 + 25 = 34\).