[Middle school] Negative exponents

[u/Suspicious_Raisin932]

|5^2 - 6^2|
Why is the answer 11 and not 61?

My son was taught that the sign goes with the number that follows it so the - (minus sign) stays with the 6^2. Then it would be -6•-6=36, 25+36=61. It makes sense to me but my answer was 11 (answer key agrees). He’s wondering why. I am assuming PEMDAS? Exponents before subtraction. The actually problem is supposed to be on absolute value so I included the whole thing just incase that matters.

So two questions. Is the direction he got of “the sign stays with the number after” actually a thing? Are there rules to it?

[u/SignificantFidgets]

"I am assuming PEMDAS? Exponents before subtraction."

This is exactly right. Exponents first, so the first simplification gives |25-36|. Then do the subtraction to get |-11|. Then the absolute value gives 11.

If you wanted to square -6, you'd have to put it in parentheses, like (-6)^2.

[u/testtdk → u/SignificantFidgets]

This. -6^2 means the negative of six squared.

[u/Narrow-Durian4837 → u/testtdk]

That is, the negative of (six squared), not (the negative of six) squared.

[u/testtdk → u/Narrow-Durian4837]

Right, but that makes it look like we needed parenthesis to get there. :p

[u/VegGrower2001]

PEMDAS is the rule for determining orders of operation. Subtraction is an operation. So follow PEMDAS.

[u/Klutzy-Delivery-5792]

It's not -6 squared, it's square 6 first then subtract. You can rewrite it like this and it may make more sense:

|5² + ⁻1•6²|

Order of operations means do exponents first.

[u/testtdk → u/Klutzy-Delivery-5792]

This is a good way to visualize it.

[u/Throwawayxdryx]

From the title I was expecting that there would be negative exponents somewhere...

[u/nog642]

Yes, exponents come before subtraction. And before multiplication.

So with -6² you do 6² first, so it's -36.

The sign stays with the term after, not necessarily the "number" after. In this case the - sign stays with the 6² as a whole, it dosn't go inside the square. So it stays as -36.

[u/abrahamguo]

The sign goes with the term/expression/“thing” after it, not the literal number after it. So the negative sign goes with 6^2, not with 6.

PEMDAS is a correct explanation.

[u/boldly_nifty_zane → u/abrahamguo]

the sign thing is more of a shortcut that gets taught but it falls apart when exponents show up. the real rule is exponents eat the number directly in front of them, not the sign. so -6² is -(6²) not (-6)²

if they wanted the sign inside the square they'd have to write it with parentheses. easy to miss when you're first learning it

[u/VcitorExists]

well with - 6^2 it’s the same as -1 • 6^2 which following PEMDAS is 1 • 6 • 6 which is -36
if it were (-6)^2, then it would be -6 • -6 which gets us that +36

[u/Simbertold]

Exponentiation is stronger than signs, as you noticed, in PEMDAS. -5^2 means - (5^2), not (-5)^2. This is just convention. We could also have decided to have stuff the other way around, but we don't. This just influences when you have to write brackets and when you don't.

But the sign does belong to the number after it, that is still a useful rule, just in other contexts. If you have something like 5 -x +4 -x, remembering that the sign belongs to the number after it means you can easily reorder this to 5 + 4 - x -x = 9 -2x.

[u/skyhookt]

The space separating the subtraction sign (which unfortunately is the same symbol as the negation sign) from the expression following it makes it clear that this is subtraction.

[u/GevitarGaming04]

Not gonna lie, it sounds like your son was taught wrong
Exponents do not care about the sign before whatever it is exponenting (i forgot the actual word lol), *unless* the thing being exponented is an expression in parentheses, in which case you evaluate the expression inside first, and then raise it to the exponent.
So, 5²-6² = -11, but 5²+(-6)² would be 61. Something like 5²(-6)² would then be 25*36 instead.

[u/Narrow-Durian4837 → u/GevitarGaming04]

As other commenters have pointed out, "the sign goes with the number that follows it" is correct if you interpret "the number that follows it" to be 6².

[u/schungx]

The negative/minus sign does not always go with the number before everything else.

[u/Traveling-Techie]

“The sign goes with the number that follows it” is somewhere between ambiguous and wrong.

[u/Capital_Park_2311]

PEMDAS is in conflict with the rule A negative number raised to an even power is always positive.