Problem 61 - College Algebra Essentials

Question

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Step-by-Step Solution

Answer

The complex conjugate of \(2+3i\) is \(2-3i\). When this complex number is multiplied by its complex conjugate, the result is the real number 13.

Step 1: Find the Complex Conjugate

The given complex number is \(2+3i\). The complex conjugate is found by changing the sign of the imaginary part. So, the complex conjugate of \(2+3i\) is \(2-3i\).

Step 2: Multiplication of Complex Number by its Conjugate

Now, we are asked to multiply the complex number \(2+3i\) by its complex conjugate \(2-3i\). You will do this as normal multiplication. Expand and simplify the result:\[(2 + 3i)(2 - 3i) = 4 - 6i + 6i - 9i^2.\]Remember that \(i^2 = -1\). So, replacing \(i^2\) with -1 gives: \[4 - 9(-1) = 4 + 9 = 13.\]