Exponent rule and square roots?

Question

For some $x$,

$\sqrt{x^2} = |x|$

However, for $x= -1$.

$\sqrt{(-1)^2} = (-1^2)^{1/2} = (-1)^{2/2} = (-1)^1 = -1$

Isn't this paradoxical?

Answer 1

To get to the problem here, we start with $\sqrt{(-1)^2}$. Please note that, before dealing with the square root, that $(-1)^2\neq -1^2$. $(-1)^2=(-1)(-1)=1$, while $-1^2=-(1^2)=-1$.

We can rewrite this using rational exponents, so $$\sqrt{(-1)^2}=((-1)^2)^{1/2}$$ I believe here, though we can just remedy the situation using our order of operations. Since we do what is in parentheses before handling exponents, we can go on and say that $$((-1)^2)^{1/2}=(1)^{1/2}=\sqrt{1}=1$$

We should also take into account the fact, though, that, as Omnomnomnomnom pointed out that we can't generally say using rules of exponents that $(a^m)^n=a^{mn}$ for all $a$. Since in this problem $a=-1$, we can't use exponential rules either.

Answer 2

As I said in the comment, the problem with this series of equalities is that we cannot generally say that $(a^{p})^q = a^{pq}$. So, in this instance, $$ ((-1)^{2})^{1/2} \neq (-1)^{2/2} $$ The equation $(a^{p})^q = a^{pq}$ will hold, however, as long when either $p$ and $q$ are both integers or $a$ is a positive number.