# How do you simplify sqrt(-32) - sqrt(-8)?

Apr 19, 2015

Since we have negative numbers under a square root operation, we obviously are in a domain of complex numbers.

Representing $- 32 = 16 \cdot 2 \cdot \left(- 1\right)$ and $- 8 = 4 \cdot 2 \cdot \left(- 1\right)$ we derive the following equivalent expression:
$\sqrt{- 32} - \sqrt{- 8} = 4 \cdot \sqrt{2} \cdot \sqrt{- 1} - 2 \cdot \sqrt{2} \cdot \sqrt{- 1}$

Now we have an interesting problem. The easy (but incomplete!) approach is replace $\sqrt{- 1}$ with complex number $i$ and derive
$4 \cdot \sqrt{2} \cdot \sqrt{- 1} - 2 \cdot \sqrt{2} \cdot \sqrt{- 1} = 4 \cdot \sqrt{2} \cdot i - 2 \cdot \sqrt{2} \cdot i = 2 \cdot \sqrt{2} \cdot i$.

It's incomplete because $\sqrt{- 1} = \pm i$. Therefore, we have two values for each member:
$4 \cdot \sqrt{2} \cdot \sqrt{- 1} = \pm 4 \cdot \sqrt{2} i$
$2 \cdot \sqrt{2} \cdot \sqrt{- 1} = \pm 2 \cdot \sqrt{2} i$
That produces four different variants of the answer:
(a) $\sqrt{- 32} - \sqrt{- 8} = 4 \cdot \sqrt{2} \cdot i - 2 \cdot \sqrt{2} \cdot i = 2 \cdot \sqrt{2} i$
(b) $\sqrt{- 32} - \sqrt{- 8} = 4 \cdot \sqrt{2} \cdot i + 2 \cdot \sqrt{2} \cdot i = 6 \cdot \sqrt{2}$
(c) $\sqrt{- 32} - \sqrt{- 8} = - 4 \cdot \sqrt{2} \cdot i - 2 \cdot \sqrt{2} \cdot i = - 6 \cdot \sqrt{2}$
(d) $\sqrt{- 32} - \sqrt{- 8} = - 4 \cdot \sqrt{2} \cdot i + 2 \cdot \sqrt{2} \cdot i = - 2 \cdot \sqrt{2}$
All four answers are equivalent and represent a possible simplification of the original expression in normal complex form.

Now the question is, how is it possible that a single expression have 4 different representations in a normal complex form. The answer is simple. The operation of square root in the domain of complex numbers has two different values. Inasmuch as $\sqrt{- 1}$ can be either $i$ or $- i$ (both, if squared, produce $- 1$), any expression that contains $\sqrt{- 1}$ has more than one representation in the complex form. We deal with an unusual type of a function - square root - that has two values for any single argument.

So, when we wright an expression $\sqrt{- 32}$ or $\sqrt{- 8}$, without additional assumption we cannot say what exactly it means, similarly to $\sqrt{- 1}$. That's why we had multitude of expressions representing the original one in a normal complex form.