Is $-1^2$ equal to 1 or -1?

Short answer: It's -1, since exponentiation takes precedence over the implied multiplication (see the section on order of operations.) Long answer: For some, that simple explanation is lacking because they argue it is the quantity -1 that is being squared, as opposed to 1 that is being squared then multiplying the result by -1. A good question to ask those making this claim is, ``Why do you associate the negative sign with just the 1, as opposed to associating the negative sign with $1^2$ ?'' Then, it usually becomes clear. Claiming the negative sign is somehow ``attached'' to just the 1, then squaring this quantity (yielding 1,) is the same as claiming the implied multiplication has a higher precedence than the exponentiation. The order of operations agreement makes clear that in fact the opposite is true--exponentiation has a higher precedence than multiplication.

Another way of interpreting $-1^2$ is $0-1^2$ . The negative sign denotes subtraction here. This is a perfectly legitimate interpretation, and the order of operations agreement tells us this is still -1 because exponentiation takes precedence over subtraction. However, it is worth noting that subtraction is usually defined in terms of addition of the negative quantity. Looking at it this way, this is really the same argument given in the previous and following interpretations.

Yet another way of interpreting the negative sign in $-1^2$ is that it denotes the unary operation of negation. In the most commonly used order of operations agreement, unary operations are not specifically mentioned since they are usually thought of as being implied by the binary operations, e.g. the unary operation of negation is thought of as being equivalent to the binary operation of multiplying by -1. It is worth mentioning that some alternative order of operations agreements actually do give unary operators higher precedence than binary operators. One example of such an agreement is the one implemented in Microsoft Excel (http://support.microsoft.com/support/kb/articles/q132/6/86.asp).

So, technically, it depends on the context (i.e. what convention is in place) as to whether $-1^2$ is 1 or -1. As explained above, using the most common order of operations agreement, $-1^2$ is perfectly unambiguous. It's -1, period. However, it is important to realize that any potential confusion should be avoided wherever it can be foreseen. On this forum, as a practical matter, it wouldn't hurt to express $-1^2$ as something like -(1^2) wherever it is feasible to do so for the simple reason that someone may find the former to be unclear. Perhaps they use Excel extensively :-) or they do not always assume what you literally type is what you literally mean (which, BTW, is often the case on alt.algebra.help).

In short, $-1^2=-1$ in most contexts, especially in the context of ``algebra,'' since most use an order of operations consistent with the one listed in this FAQ, unless advised explicitly on a case-by-case basis that a different agreement is in use.