# A small exercise

When calculations are performed, people often use several arithmetic and algebraic rules without thinking about how these rules *really* work. The following example illustrates this fact:

So: **1 equals -1**.

Apparently at least one of the equal signs above must be wrong. But which one? Or which ones?

I have noticed that many high school teachers find this problem difficult to solve. And that says more about the teachers than the problem!

((-1)²)^(1/2)) can’t be reduced to (-1)^(2*(1/2)), it’s not commutative … it would be akin to say that sqrt ((-1)²)=1 equals (sqrt(-1))²=i²=-1

That is correct, visitor Pelele!!!

The rule

is(a^b)^c = a^(bc)notvalid whenis negative, which some people tend to forget.a(But it is always valid when

is greater than zero.)a