Express each of the following in the form (a + ib):

Solution:

Given: $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$

Now, rationalizing

$=\frac{5+\sqrt{2} i}{1-\sqrt{2} i} \times \frac{1+\sqrt{2} i}{1+\sqrt{2} i}$

$=\frac{(5+\sqrt{2} i)(1+\sqrt{2} i)}{(1-\sqrt{2} i)(1+\sqrt{2} i)} \ldots$ (i)

Now, we know that,

$(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

So, eq. (i) become

$=\frac{(5+\sqrt{2} i)(1+\sqrt{2} i)}{(1)^{2}-(\sqrt{2} i)^{2}}$

$=\frac{5(1)+5(\sqrt{2} i)+\sqrt{2} i(1)+\sqrt{2} i(\sqrt{2} i)}{1-2 i^{2}}$

$=\frac{5+5 \sqrt{2} i+\sqrt{2} i+2 i^{2}}{1-2(-1)}\left[\because \mathrm{i}^{2}=-1\right]$

$=\frac{5+6 i \sqrt{2}+2(-1)}{1+2}$

$=\frac{3+6 i \sqrt{2}}{3}$

$=\frac{3(1+2 i \sqrt{2})}{3}$

$=1+2 \mathrm{i} \sqrt{2}$