Where the Minus Lives

These two look like twins and are not: (-5)^2 and -5^2. In the first, the parentheses put the minus inside the base, so both factors are negative: (-5)(-5) = 25. In the second there are no parentheses, and the convention is that the exponent binds first: -5^2 means "the opposite of 5^2", which is -25.

The rule of the road: an exponent only reaches what it is directly attached to. Attached to a parenthesis, it powers everything inside. Attached to a bare 5, it powers the 5 and nothing else, and the minus waits outside until the power is done.

Fifty points on an exam have been lost to this convention. When you mean to square a negative number, write the parentheses. When you read an expression without them, let the exponent go first.