William Montgomery

2022-02-02

How do you divide $\frac{2i}{4-5i}?$

KickAntitte06

Expert

Step 1
Multiply by the complex conjugate.
$\frac{2i}{4-5i}\left(\frac{4+5i}{4+5i}\right)=\frac{8i+10{i}^{2}}{16+20i-20i-25{i}^{2}}=\frac{8i+10{i}^{2}}{16-25{i}^{2}}$
Recall that $i=\sqrt{-1}$, so ${i}^{2}=-1$
$\frac{8i+10\left(-1\right)}{16-25\left(-1\right)}=\frac{8i-10}{16+25}=\frac{-10+8i}{41}$
In the form of a complex number, this would be written as:
$-\frac{10}{41}+\frac{8}{41}i$

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