How do you simplify (5-isqrt3)/(5+isqrt3)?

Oct 27, 2016

The simplification $= \frac{11}{14} - \frac{i 5 \sqrt{3}}{14}$

Explanation:

To simplify a complex numbers, we must multiply by the conjugate of the denominator
if $z = {z}_{1} / {z}_{2}$ then $z = \frac{{z}_{1} {\overline{z}}_{2}}{{z}_{2} {\overline{z}}_{2}}$

In our case ${\overline{z}}_{2} = 5 - i \sqrt{3}$
${i}^{2} = - 1$

so $\frac{\left(5 - i \sqrt{3}\right) \left(5 - i \sqrt{3}\right)}{\left(5 + i \sqrt{3}\right) \left(5 - i \sqrt{3}\right)} = \frac{25 - 10 i \sqrt{3} - 3}{25 + 3}$

$= \frac{22 - 10 i \sqrt{3}}{28} = \frac{11}{14} - \frac{i 5 \sqrt{3}}{14}$