# What is the conjugate of sqrt(-20)?

Nov 21, 2016

$- 2 \sqrt{5} i$

#### Explanation:

Given a complex number $z = a + b i$ (where $a , b \in \mathbb{R}$ and $i = \sqrt{- 1}$), the complex conjugate or conjugate of $z$, denoted $\overline{z}$ or ${z}^{\text{*}}$, is given by $\overline{z} = a - b i$.

Given a real number $x \ge 0$, we have $\sqrt{- x} = \sqrt{x} i$.

note that ${\left(\sqrt{x} i\right)}^{2} = {\left(\sqrt{x}\right)}^{2} \cdot {i}^{2} = x \cdot - 1 = - x$

Putting these facts together, we have the conjugate of $\sqrt{- 20}$ as

$\overline{\sqrt{- 20}} = \overline{\sqrt{20} i}$

$= \overline{0 + \sqrt{20} i}$

$= 0 - \sqrt{20} i$

$= - \sqrt{20} i$

$= - 2 \sqrt{5} i$