Krdytkyu

A few people I met were debating over if $-5^2 = 25$ or $-25$.

From my experience, we assume operator precedence and get $-25.$
People are telling me however, calculators are flawed, the real answer is $25.$

Clarifying note, the expression is merely:
$-5^2$
nothing else.
What would you say?

Edit:
I've also tried saying $f(x)=-x^2$ is a parabola that opens downwards, but this was apparently a problem with graphing calculators as well.

By definition of the order of operations, exponentiation takes precedence over negation. Therefore,
$$
-5^2 = (-1) times 5 times 5 = -25
$$
but the alternative would be
$$
(-5)^2 = left[(-1) times 5right] times left[(-1) times 5 right] = 25.
$$

You’re right. $-5^2$ means $-1cdot 5^2$.

$$-5^2 = -1cdot 5^2 = -25$$

They’re confusing it with $(-5)^2$, which means $(-5)cdot (-5)$.

$$(-5)^2 = (-5)cdot (-5) = +25$$

When -5^2 is written on paper, it's assumed that it's actually (-5)^2, as in the negative was already applied to the 5 before squaring. Often times this is where the confusion comes from when typing -5^2 into a calculator is equal to -25.
The calculator is not necessarily wrong. When you type -5^2, the calculator is really calculating -(5)^2. It squares the 5 first, then it applies the negative(multiplies the equation by negative 1.) This is really what the negative sign means, multiplying the number by -1. When we teach and learn -'s though, we automatically multiply it out, since -5 and -(5) equals the same thing, which is -5. Just know that when you put a negative sign in your calculator, you're really putting "-1x...", "negative one times..."

Without a parentheses the negative is taken as a -1, so it would read as ${-1*5^2}$, which would equal -25. With a parentheses, it would be read as $({-5)^2}$ which equals 25.

To properly answer you question, I need a graphing calculator. I don't have one, but I do have a scientific calculator, a CVS knock-off of a Sharp 2-line model.

I pressed the negation key "+/-", the display shows "-0". So far so good. Then I press 5 and the display changes to "-5". But then I press the $x^2$ key and then "(-5)²" goes up on the top line of the display. So this calculator has clearly understood my button pushing to mean "$(-5)^2$", not "$-(5^2)$".

Hmm... I wonder about reverse Polish... $(-5)^2$ would be $5 - 5 - times$, while $-(5^2)$ would be $0 , 5 , 5 times -$ (press Enter when you need to follow a number with another number).

If Jerry Springer got involved in this debate, I think he would cap it off saying that "Ultimately, communication among humans is subject to the vagaries of shared experiences, and in those gaps, ambiguities creep in." Or something along those lines.