$(-x)^2$ is definately equal to $(-1)^2(x)^2$, right?
But $-(x)^2$ and $-x^2$ are confusing me, do they mean $-(x^2)$ or do they mean $(-1)^2(x)^2$?
$\endgroup$ 03 Answers
$\begingroup$Because the unary $-$ has lower priority than exponentiation, convention$^\ast$ says
$$-(x)^2 = -x^2 = -(x^2) = -x\cdot x$$
The parentheses just clarify the meaning here. In the expression $(-x)^2 = (-x)\cdot(-x) = x^2$ they are essential.
This convention is very similar to the way that
$$3-2\cdot 4 = 3 - (2\cdot 4)$$
must be handled.
$^\ast$ some programming languages have a different convention (for example Perl), but generally this is the standard unless stated otherwise, such as in a language specification. See this wikipedia section on exceptions
$\endgroup$ 7 $\begingroup$$(-x)^2=(-x)(-x)=(-1)^2(x)^2=x^2$
$-(x)^2=-(x\cdot x)=-x^2$
$-x^2=-(x^2)$ as shown above
$\endgroup$ $\begingroup$It depends on the medium you are using. Given that exponentiation is evaluated before subtraction by convention (probably from the study of polynomials), there become 2 competing design decisions:
- That subtraction and negation are evaluated with the same priority
- That unary operators are universally evaluated before binary operators
Most academic publications will use the first convention, that $-x^2 = -(x^2)$.
A lot of software (Haskell, some spreadsheet software) will use the second convention so $-x^2 = (-x)^2$.
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