confused about law of exponent (contradiction) Is the following statement true?
$$
(-1.2)^{1.6} = \sqrt[10]{(-1.2)^{16}}.
$$  
Is it because $a^{0.5} = a^{1/2} = \sqrt{a}$?
 A: I'll take $\sqrt{}$ as example because for odd numbers, you get a bijection from $\mathbb{R}$ into itself (with positive number mapped to positive numbers and negative numbers mapped to negative numbers) so the problem only arises with even numbers and once you understand for $\sqrt{}$, you'll understand for $\sqrt[4]{}$, $\sqrt[6]{}$ etc. as well.
You problem is that you didn't understand the function $\sqrt{} : \mathbb{R_+} \to  \mathbb{R_+}$ is just a convention.
When we write $\sqrt{4}$, we mean $2$ because we chose the positive real number as result for the function $\sqrt{}$ because if we didn't specify that, we wouldn't get a function since a function is something that has a unique value for each "input".
But if you try to extend that function to $\mathbb{R}$ we get $\sqrt{} : \mathbb{R} \to ?$ where does it go? Because now, for $\sqrt{-1}$, you have to decide between $i$ and $-i$...
You would pick $i$ because it looks positive but it is not... There has been a question about a way to differentiate $i$ from $-i$ recently on SO and the answers were, more or less, that you can't.
Of course, for $\mathbb{R}$, you could decide to always take the one that is a positive real number multiplied by $i$. But then you get $\sqrt{} : \mathbb{R} \to \mathbb{C}$ and you only use $i$ in $\mathbb{C}$ so you don't get a bijection.
And then if you try to extend it to $\mathbb{C}$ with $\sqrt{} : \mathbb{C} \to ?$ you can't decide which one to take unless you define some kind of sign over $\mathbb{C}$ to partition it. Which you usually don't want to do.
Plus when you work in $\mathbb{C}$, you usually just want one of the answers, of $a^2=b$ and you don't care which one you get.
A: First, take a look at this Wikipedia article.
The problem here is that the base number $-1.2$ is negative. Some people think of negative numbers being raised to non-integer powers as being undefined. However, one might reasonably think of for example $(-1)^{\frac{1}{3}}$ as the cube root of $-1$, which is $-1$. It might be a bit confusing, but I think if you look at the Wikipedia article that I linked to above, you might find a good answer.
You have a rational exponent $1.6 = \frac{16}{10} = \frac{8}{5}$ (reduced fraction). $5$ is odd you have something well-defined and you get 
$$
(-1.2)^{\frac{8}{5}} = \sqrt[5]{(-1.2)^8}.
$$
This is equal to what you have written.
Now, you might have a calculator that doesn't evaluate the left hand side. I just tried with my calculator (TI-84) and had no problem.
