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# Help!

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The complex numbers a and b satisfy a $$\overline{b} = -1 + 5i.$$ Find $$\overline{a} b.$$

Apr 16, 2019

#1
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Set $$a=c+di$$ and $$b=r+si$$ where $$c$$$$d$$$$r$$, and $$s$$ are real numbers. This makes the $$a\overline{b}=-1+5i$$ turn into $$(c+di)(r-si)=-1+5i$$. When we multiply the left side out, we get $$(cr+ds)+(dr-cs)i$$. So, $$cr+ds=-1$$ and $$dr-cs=5$$.

Notice that $$\overline{a}b=(c-di)(r+si)=(cr+ds)+(cs-dr)i=(cr+ds)-(dr-cs)i$$. Plugging in the values before gives us $$\overline{a}b=-1-5i$$.

Hope this helps!

Apr 16, 2019
#2
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The complex numbers $$a$$ and $$b$$ satisfy $$a\overline{b} = -1 + 5i$$.
Find $$\overline{a} b$$.

$$\begin{array}{|rcll|} \hline a\overline{b} &=& -1 + 5i \\\\ \overline{a\overline{b}} &=& \overline{-1 + 5i} \quad | \quad \overline{\overline{b}} = b \\\\ \overline{a} b &=& \overline{-1 + 5i} \quad | \quad \overline{-1 + 5i} = -1-5i \\\\ \mathbf{\overline{a} b} &\mathbf{=}& \mathbf{-1-5i} \\ \hline \end{array}$$

Apr 17, 2019