If $(a + i b) (c + i d) (e + i f) (g + i h) = A + i B$ , then prove that $(a^{2} + b^{2}) (c^{2} + d^{2}) (e^{2} + f^{2}) (g^{2} + h^{2}) = A^{2} + B^{2}$ .

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Solution

$(a + i b) (c + i d) (e + i f) (g + i h) = A + i B$
$∴ | (a + i b) (c + i d) (e + i f) (g + i h) | = | A + i B |$
or $| (a + i b) | \times | (c + i d) | \times | (e + i f) | \times | (g + i h) | = | A + i B |$
or $\sqrt{a^{2} + b^{2}} \times \sqrt{c^{2} + d^{2}} \times \sqrt{e^{2} + f^{2}} \times \sqrt{g^{2} + h^{2}} = \sqrt{A^{2} + B^{2}}$
On squaring both sides, we get
$(a^{2} + b^{2}) (c^{2} + d^{2}) (e^{2} + f^{2}) (g^{2} + h^{2}) = A^{2} + B^{2}$

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