Question

The simplest form of the expression $\frac{10-\sqrt{-12}}{1-\sqrt{-27}}$ is

• A. $-\frac{2}{7}$
• B. $\frac{28}{3}$
• C. $-\frac{2}{7}+i\sqrt{3}$
• D. $1+i\sqrt{3}$

Answer 1

The correct option is D $1+i\sqrt{3}$

We need to find simplest form of $\frac{10-\sqrt{-12}}{1-\sqrt{-27}}$

$=\frac{10-2i\sqrt{3}}{1-3i\sqrt{3}}$

Taking conjugate, we get

$=$ $\frac{10-2\sqrt{-3}}{1-3\sqrt{-3}}\times \frac{1+3i\sqrt{3}}{1+3i\sqrt{3}}$

$=$ $\frac{\frac{10-2\sqrt{-3}}{1-3\sqrt{-3}}}{1^2-(3i\sqrt{3}{)}^2}$

$=$ $\frac{10+30i\sqrt{3}-2i\sqrt{3}-18i^2}{28}$

$=$ $\frac{10+28i\sqrt{3}+18}{28}$

$=$ $\frac{28(1+i\sqrt{3})}{28}$

$=$ $1+i\sqrt{3}$