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I always had this confusion of when I need to apply the negative sign in the calculation. I understand that $(-1)^2 = 1$ however why isn't $-1^2 = 1$?

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    $\begingroup$ because $(-1)^2=(-1)*(-1)=1$, but $-1^2 =-(1^2)=-(1*1)=-(1)=-1$ $\endgroup$ – Luke Apr 21 at 22:01
  • $\begingroup$ Though beware Excel and some similar cases, where =-1^2 gives 1 but =0-1^2 gives -1, because if interprets the former as $(-1)^2$ and the latter as $0-(1^2)$, i.e. the first - as a unary operation taking precedence over exponentiation and the second - as a binary operation with exponentiation taking precedence over it $\endgroup$ – Henry Apr 21 at 22:14
  • $\begingroup$ Just for an example, that's the same as writing $-1 \times 1^2 = 1$, which probably is pretty clear that it's not true $\endgroup$ – MCMastery Apr 21 at 23:40
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When we write $-x^2$, it means we square $x$ first, then take the negative of this. That is, $$-x^2 = -\left(x^2\right).$$ So $$-1^2 = -\left(1^2\right)=-1.$$ (And thus $-x^2$ means something different to $(-x)^2$.)

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Unary minus has lower precedence than elevation to a power.

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As it is already in the previous answers:
$(-x)^2\neq-x^2$ To avoid confusion, it is better to use parentheses. $-x^2=-(x^2)$

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