What is expand i/((a+bi)^2) ?

The solution to expand i/((a+bi)^2) is (2ab)/(a^4+b^4+2a^2b^2)+i(a^2-b^2)/(a^4+b^4+2a^2b^2)

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expand i/((a+bi)^2)

expand

\frac{i}{( a + bi ) ^{2}}

\frac{2 ab}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}} + i \frac{a ^{2} - b ^{2}}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}}

Solution steps

\frac{i}{( a + bi ) ^{2}}

(a + bi)^{2} = a^{2} + 2 iab - b^{2}

= \frac{i}{a ^{2} + 2 iab - b ^{2}}

Rationalize

\frac{i}{a ^{2} + 2 iab - b ^{2}} : \frac{2 ab + i ( a ^{2} - b ^{2} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

= \frac{2 ab + i ( a ^{2} - b ^{2} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

Rewrite

\frac{2 ab + i ( a ^{2} - b ^{2} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

in standard complex form:

\frac{2 ab}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{a ^{2} - b ^{2}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

= \frac{2 ab}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{a ^{2} - b ^{2}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

\frac{2 x ^{3} + 5 x - 1}{( 4 x + 7 ) ^{2} ( x ^{2} + 1 ) ^{3}}

(x^{2} + e^{4 x} - 7)^{2}

ro t (x^{2} y, 1 + y, x z^{2})

\frac{2 ( 1 + x ) ^{3}}{3}

(3 + 3 i - 4 i t) e^{- 1 + i}

What is expand i/((a+bi)^2) ?
The solution to expand i/((a+bi)^2) is (2ab)/(a^4+b^4+2a^2b^2)+i(a^2-b^2)/(a^4+b^4+2a^2b^2)