What is expand i/((1+2sqrt(3)i)^5) ?

The solution to expand i/((1+2sqrt(3)i)^5) is (58sqrt(3))/(371293)+i 601/371293

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expand i/((1+2sqrt(3)i)^5)

expand

\frac{i}{( 1 + 2 3 i ) ^{5}}

\frac{58 3}{371293} + i \frac{601}{371293}

Solution steps

\frac{i}{( 1 + 2 3 i ) ^{5}}

(1 + 23 i)^{5} = 601 + 583 i

= \frac{i}{601 + 58 3 i}

Rationalize

\frac{i}{601 + 58 3 i} : \frac{58 3 + 601 i}{371293}

= \frac{58 3 + 601 i}{371293}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{58 3 + 601 i}{371293} = \frac{58 3}{371293} + \frac{601 i}{371293}

= \frac{58 3}{371293} + \frac{601 i}{371293}

Rewrite

\frac{58 3}{371293} + \frac{601 i}{371293}

in standard complex form:

\frac{58 3}{371293} + \frac{601}{371293} i

= \frac{58 3}{371293} + \frac{601}{371293} i

(2 x y + z^{2}) d x + (2 yz + x^{2}) d y + (2 x z + y^{2}) d z

\frac{5 z - 5 x}{( z + 3 ) ( x + 3 )}

12 (2 - 2) + 54

\frac{1}{( x + 4 ) ( x + 5 )}

\frac{1}{x} a t (- 2, - \frac{1}{2})

What is expand i/((1+2sqrt(3)i)^5) ?
The solution to expand i/((1+2sqrt(3)i)^5) is (58sqrt(3))/(371293)+i 601/371293