# what does $-2^2$ evaluate to?

What does $-2^2$ evaluate to? My wife is upgrading her math & I'm quite uncertain how to answer.

is it the same as $(-2)^2$ or is it the same as $-(2^2)$?

at the heart of this is whether the minus sign is associated with the 2 or the expression $2^2$.

thank you! mp

• $-2^3 = -(2^3)$ – The Chaz 2.0 Sep 30 '11 at 18:15
• The usual convention is that exponentiation occurs first, multiplication by constants and negative signs occur later. If you are following the usual conventions, then $-2^2$ would be interpreted as "do $2^2$ first, then multiply the answer by $-1$", to get $-4$. However, this is a convention. In some contexts, it might have been agreed that $-2^2$ means $(-2)^2$. – Arturo Magidin Sep 30 '11 at 18:18
• thank you TheChaz--corrected the title. thank you arturo for setting the super script...didn't see how to do that. your reply is consistant with the book's chapter..."orders of operation". thank you all. – x-ray Sep 30 '11 at 18:22

There are different conventions. In mathematics the usual one is to write $(-2)^2$ if you want to raise a negative number to a power, whereas $-2^2$ usually means $-(2^2)$.
This is, however, not completely universal. For example, many programming languages (that otherwise follow mathematical conventions for arithmetic expressions) consider a unary minus to bind tighter than any binary operator. In such languages, -2**2 (or whatever their syntax for exponentation is) would mean (-2)**2.
• I think the convention is universal in mathematics, if not within programming languages. I normally either comment or deduct points, saying it's incorrect, if I'm grading something where a student wrote $-2^2$ while clearly intending $(-2)^2$ or where $(-2)^2$ is correct. – Michael Hardy Sep 30 '11 at 19:12
• Actually, the C-based languages do not have an exponentiation operator, but use functions instead, so there would be no ambiguity. Fortran uses ** but it places that ahead of negation in precedence, and I would imagine that other languages styled after Fortran use the same precedence. – Daniel R Hicks Oct 1 '11 at 3:59
• python reports $-2**2 == -4$ as well, and I'm virtually certain magma and mathematica agree. Since you've mentioned fortran, that completely exhausts the languages I have at least passing familiarity with that I know have a binary exponentiation operator. – Hurkyl Oct 1 '11 at 16:36