Question

Express the given equation in the form of a + ib, a, b ∈ R $i=\sqrt{-1}$. State the values of a and b.
$\frac{2+\sqrt{-3}}{4+\sqrt{-3}}$

Answer 1

As we know that

$\sqrt{ - 1} = i \\ \\ {i}^{2} = - 1$
$\frac{2+\sqrt{-3}}{4+\sqrt{-3}} \\ \\ = \frac{2+i\sqrt{3}}{4+i\sqrt{3}} \times \frac{4 - i \sqrt{3} }{4 - i \sqrt{3} } \\ \\ = \frac{8 - 2i \sqrt{3} + 4i \sqrt{3} - 3 {i}^{2} }{( {4)}^{2} - ( {i \sqrt{3}) }^{2} } \\ \\ = \frac{8 + 3 + 2i \sqrt{3} }{16 + 3} \\ \\ = \frac{11 + 2i \sqrt{3} }{19} \\ \\ a + ib = \frac{11}{19} + i \frac{2 \sqrt{3} }{19} \\ \\ a = \frac{11}{19} \\ \\ b = \frac{2 \sqrt{3} }{19}$
Hope it helps you.