Question 233405
The key difference between your two examples is that in {{{2^(-2)}}} the minus sign is on the exponent and in {{{(-2^3)^2}}} the minus sign is not in an exponent. A minus sign in an exponent means "reciprocal of". It does <b>not</b> mean that a negative number is being multiplied anywhere.
P.S. {{{(-2^3)^2 = (-(2^3))^2 = (-(2*2*2))^2 = (-8)^2 = 64}}}. The only way to get {{{2^6}}} while simplifying is an unusual path:
{{{(-2^3)^2 = (-(2^3))^2 = ((-1^3)(2^3))^2 = ((-1)^6)(2^6) = 2^6 = 64}}}
The important point here is that {{{-2^3}}} means {{{-(2*2*2)}}}. The exponent applies only to the 2, not to the minus sign. On the other hand, {{{(-2)^3}}} means {{{(-2)(-2)(-2)}}}. The difference may seem trivial because with an odd exponent like 3 the two work out the same. But with even exponents, the difference is critical.<br>
<b>Exponents apply only to what is immediately in front of them.</b> In the first case, the first thing in front of the 3 is 2. In the second case, it is ")" in front of the 3 (which means the exponent applies to the entire expression in the parentheses).