How do you simplify sqrt(-3) + sqrt(-27)?

Nov 27, 2015

$\sqrt{- 3} + \sqrt{- 27} = 4 \sqrt{3} i$

Explanation:

If $x < 0$ then $\sqrt{x} = \left(\sqrt{- x}\right) i$

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

So:

$\sqrt{- 3} + \sqrt{- 27}$

$= \left(\sqrt{3}\right) i + \left(\sqrt{27}\right) i$

$= \left(\sqrt{3} + \sqrt{27}\right) i$

$= \left(\sqrt{3} + \sqrt{{3}^{2} \cdot 3}\right) i$

$= \left(\sqrt{3} + \sqrt{{3}^{2}} \sqrt{3}\right) i$

$= \left(1 \sqrt{3} + 3 \sqrt{3}\right) i$

$= 4 \sqrt{3} i$