# How do you simplify 12sqrt(-2)^12?

Mar 21, 2016

$12 \sqrt{{\left(- 2\right)}^{12}} = 3 \cdot {2}^{8} = 768$

or

$12 \sqrt{- {2}^{12}} = 768 i \in \mathbb{C}$

#### Explanation:

in $\mathbb{R}$ the expression you writed has no semplification, but if you intend:

$12 \sqrt{- {2}^{12}} = 12 \sqrt{{\left(- 2\right)}^{12}}$

You can rewrite expression as:

$12 \sqrt{{\left(- 1\right)}^{12} \cdot {2}^{12}} = 12 \sqrt{{2}^{12}} = 12 \sqrt{{\left({2}^{6}\right)}^{2}} = 12 \cdot {2}^{6} = 3 \cdot {2}^{2} \cdot {2}^{6} = 3 \cdot {2}^{8} = 768$

Indeed, that's different from

$12 \sqrt{- {2}^{12}} = 12 \sqrt{- 1} \cdot \sqrt{{2}^{12}} = 12 \cdot i \cdot \sqrt{{\left({2}^{6}\right)}^{2}} = 768 i$

with $i = \sqrt{- 1}$ as immaginary unit of $\mathbb{C}$