Suppose $(a+bi) (c+di)(e+di) = 3+8i$ where $a,b,c,d,e,f$ are real number find the value $(a^2+b^2)(c^2+d^2)(e^2+f^2)$

Question

[Suppose $(a+bi)(c+di)(e+di) = 3+8i$ where $a,b,c,d,e,f$ are real number find the value $(a^2+b^2)(c^2+d^2)(e^2+f^2)$

Answer 1

Note that $$(a-bi)(c-di)(e-fi) = 3-8i$$ and also that $$a^2+b^2 = (a-bi)(a+bi)$$

Hence $$(a-bi)(c-di)(e-fi) \cdot (a+bi)(c+di)(e+fi) = (3-8i) \cdot (3+8i)$$ $$\implies (a^2+b^2)(c^2+d^2)(e^2+f^2) = (3-8i)(3+8i)$$