SOLUTION: Regarding negative exponents. Please explain why 2^-2 evaluates to 1/2^2 =1/4, but (-2^3)^2 evaluates to 2^6=64. I do not understand why the second expression results in a whole

Algebra ->  Algebra  -> Exponents-negative-and-fractional -> SOLUTION: Regarding negative exponents. Please explain why 2^-2 evaluates to 1/2^2 =1/4, but (-2^3)^2 evaluates to 2^6=64. I do not understand why the second expression results in a whole       Log On

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Question 233405: Regarding negative exponents. Please explain why 2^-2 evaluates to 1/2^2 =1/4, but (-2^3)^2 evaluates to 2^6=64. I do not understand why the second expression results in a whole number. bljones1944@charter.net
Answer by jsmallt9(2063) About Me  (Show Source):
You can put this solution on YOUR website!
The key difference between your two examples is that in 2%5E%28-2%29 the minus sign is on the exponent and in %28-2%5E3%29%5E2 the minus sign is not in an exponent. A minus sign in an exponent means "reciprocal of". It does not mean that a negative number is being multiplied anywhere.
P.S. %28-2%5E3%29%5E2+=+%28-%282%5E3%29%29%5E2+=+%28-%282%2A2%2A2%29%29%5E2+=+%28-8%29%5E2+=+64. The only way to get 2%5E6 while simplifying is an unusual path:
%28-2%5E3%29%5E2+=+%28-%282%5E3%29%29%5E2+=+%28%28-1%5E3%29%282%5E3%29%29%5E2+=+%28%28-1%29%5E6%29%282%5E6%29+=+2%5E6+=+64
The important point here is that -2%5E3 means -%282%2A2%2A2%29. The exponent applies only to the 2, not to the minus sign. On the other hand, %28-2%29%5E3 means %28-2%29%28-2%29%28-2%29. 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.

Exponents apply only to what is immediately in front of them. 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).