Variable power with negative irrational exponent, wrong result but why?

Question

I'm sorry for my english math terms, I'm trying to translate them from Italian, hopefully everything is correct.

On my textbook I have this exercise (Powers with irrational exponents chapter):

$(a^2-1)^{-\sqrt{2}}$

And I'm asked to say for what values of $a$ this power is meaningful so this is what I did:

I know that $a^2 -1$ must be greater than $0$ because the exponent $-\sqrt{2}$ is a negative irrational number

$a^2-1>0$

$a^2>1$

$a>\pm\sqrt1$

$a>\pm1$

Now I know this is wrong but I really have no idea how to get to the correct result which is:

$a<-1\lor a>1$

I've searched on the book, online but never received any satfisying answer.

Answer 1

You have to be very careful when taking square roots because $\sqrt{a^2}$ really means $|a|$. Your confusion arises when you claim that $a > \pm \sqrt{1}$ because, while this is true for equalities, it isn't necessarily true for inequalities.

In this case you have two cases; either $a$ is positive or $a$ is negative. If $a$ is positive then we can conclude that $a > 1$. However, if $a$ is negative then we have to be careful about our signs: $-a > 1 \implies a < -1$.

A safer way would be to factor the original expression as a difference of squares:

\begin{align} a^2 - 1 &> 0 \\ (a-1)(a+1) &> 0, \end{align}

and then you have $3$ regions to check.

Answer 2

After $a^2 > 1$ you can conclude with $a<-1 \lor a>1$. [Think about which numbers $a$, after squaring, become $>1$.] Be sure you understand why going to $a>\pm \sqrt{1}$ is wrong.