Problem

Given the complex number \(z = 4 + 3i\), find the complex conjugate of \(z\) and then multiply \(z\) with its complex conjugate.

Answer

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Answer

Step 4: Simplify the expression. The \(-12i + 12i\) cancels out, leaving \(16 - 9 = 7\).

Steps

Step 1 ：Step 1: Find the complex conjugate of \(z\). The complex conjugate of a complex number \(a + bi\) is \(a - bi\). So, the complex conjugate of \(4 + 3i\) is \(4 - 3i\).

Step 2 ：Step 2: Multiply \(z\) with its complex conjugate. To do this, we need to multiply \((4 + 3i)(4 - 3i)\).

Step 3 ：Step 3: Using the distributive property, we expand this to \(4*4 + 4*(-3i) + 3i*4 + 3i*(-3i) = 16 - 12i + 12i - 9\).

Step 4 ：Step 4: Simplify the expression. The \(-12i + 12i\) cancels out, leaving \(16 - 9 = 7\).

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