X-iy- a-ib c-id - prove that x 2 - y 2 2 - a 2 - b 2 c 2 - d 2-

Given x+ iy = √(a + i bc+ id) ___(1)We Replace i by i x iy = √(a ibc id) ____(2)multiply equation (1) amp; (2) (x + i ...

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Algebra of Complex Numbers

x+iy=√ a+ib c...

Question

$x + i y = \sqrt{\frac{a + i b}{c + i d}}$ , prove that ${(x^{2} + y^{2})}^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$ .

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Solution

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x - i y = \sqrt{\frac{a - i b}{c - i d}}

{(x^{2} + y^{2})}^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

(x + i y) = \sqrt{\frac{a + i b}{c + i d}}

(x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

If $x - i y = \sqrt{\frac{a - i d}{c - i d}},$ prove that $(x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$ .

\frac{(a + i b)}{(c + i d)} = x + i y

\frac{a - i b}{c - i d} = x - i y

\frac{a^{2} + b^{2}}{c^{2} + d^{2}} = x^{2} + y^{2}

I f x + i y = \sqrt{\frac{a + i b}{c + i d}}, t h e n (x^{2} + y^{2})^{2}

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