Add, subtract, and multiply complex numbers using i^2=-1 and algebraic properties.
Multiply a complex number by its conjugate
Problem
Recognize \((3+4i)(3-4i)\) as a conjugate product. Apply \((a+\text{bi})(a-\text{bi})=a^{2}+b^{2}\) and calculate the real value.
Big Picture
What this problem is really about
Conjugate factors share a real part and have opposite imaginary parts, causing the cross terms to cancel. We’ll recognize that structure, apply the sum-of-squares product identity created by the imaginary square, evaluate both squares, and confirm that no imaginary component remains.
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