Question

multiply 5 - 6i by its complex conjugate.
the product is \boxed{}

Explanation:

Step1: Find the complex conjugate

The complex conjugate of \(a - bi\) is \(a + bi\). For \(5 - 6i\), its complex conjugate is \(5 + 6i\).

Step2: Multiply the two complex numbers

Use the formula \((a - bi)(a + bi)=a^2 + b^2\) (since \((a - bi)(a + bi)=a^2 - (bi)^2=a^2 - b^2i^2=a^2 + b^2\) as \(i^2 = - 1\)). Here, \(a = 5\) and \(b = 6\). So, \((5 - 6i)(5 + 6i)=5^2+(6)^2\).
Calculate \(5^2 = 25\) and \(6^2=36\). Then \(25 + 36=61\).

Answer:

\(61\)