Simplify each of the following and express it in the form a + ib
Question:

Simplify each of the following and express it in the form a + ib

$(5+\sqrt{-3})(5-\sqrt{-3})$

Solution:

Given: $(5+\sqrt{-3})(5-\sqrt{-3})$

We re – write the above equation

$(5+\sqrt{(-1) \times 3})(5-\sqrt{(-1) \times 3})$

$=\left(5+\sqrt{3 i^{2}}\right)\left(5-\sqrt{3 i^{2}}\right)\left(\because, i^{2}=-1\right]$

$=(5+i \sqrt{3})(5-i \sqrt{3})$

Now, we know that,

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

Here, $a=5$ and $b=i \sqrt{3}$

$=(5)^{2}-(i \sqrt{3})^{2}$

$=25-\left(3 i^{2}\right)$

$=25-[3 \times(-1)]$

$=25+3$

$=28+0$

$=28+0 i$