Question 120436
In order to simplify this, we need to get everything into standard form.


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*[Tex \LARGE \frac{1}{9+2i}] Start with the first part of the expression.


*[Tex \LARGE \left(\frac{1}{9+2i}\right)\left(\frac{9-2i}{9-2i}\right)] Multiply the fraction by *[Tex \LARGE \frac{9-2i}{9-2i}]


*[Tex \LARGE \frac{9-2i}{85}] Foil and Multiply



*[Tex \LARGE \frac{9}{85}-\frac{2}{85}i] Break up the fraction. So it is now in {{{a+bi}}} form where {{{a=9/85}}} and {{{b=-2/85}}}


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*[Tex \LARGE \frac{1}{2+1i}] Start with the second part of the expression


*[Tex \LARGE \left(\frac{1}{2+1i}\right)\left(\frac{2-1i}{2-1i}\right)] Multiply the fraction by *[Tex \LARGE \frac{2-1i}{2-1i}]


*[Tex \LARGE \frac{2-1i}{5}] Foil and Multiply



*[Tex \LARGE \frac{2}{5}-\frac{1}{5}i] Break up the fraction. So it is now in {{{a+bi}}} form where {{{a=2/5}}} and {{{b=-1/5}}}




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So we the expression goes from *[Tex \LARGE \frac{1}{9 + 2i} - \frac{1}{2+i}] and becomes:


*[Tex \LARGE \frac{9}{85}-\frac{2}{85}i-\left(\frac{2}{5}-\frac{1}{5}i\right)]




*[Tex \LARGE \frac{9}{85}-\frac{2}{85}i-\frac{2}{5}+\frac{1}{5}i] Distribute the negative



*[Tex \LARGE \left(\frac{9}{85}-\frac{2}{5}\right)+\left(-\frac{2}{85}i+\frac{1}{5}i\right)] Group like terms



*[Tex \LARGE -\frac{5}{17}+\frac{3}{17}i] Combine like terms




So *[Tex \LARGE \frac{1}{9 + 2i} - \frac{1}{2+i}] simplifies to *[Tex \LARGE -\frac{5}{17}+\frac{3}{17}i] 




In other words *[Tex \LARGE \frac{1}{9 + 2i} - \frac{1}{2+i}=-\frac{5}{17}+\frac{3}{17}i]