Daughter's math homework (1 Viewer)

[lombardi]

This alludes back to an earlier post, but what would your answers be for these?

4 x 2³ - 5² =

2 x 3² - 6² =

4*8-25= 7But, had it been presented as -5^2 + 4*2^3 I would've said 25+32=57

But the two should be equal:4 x 2³ - 5² = -5² + 4 x 2³

Left side is 7. Right side is 57.NO, the 2 are not equal.

As mentioned in this earlier post by Iguana, they should be:

The answer lies in the commutative property of addition.

Given that a + b = b + a:

-5² + 4 x 2³ = 4 x 2³ + -5²

Now it should be obvious that:

4 x 2³ + -5² = 4 x 2³ - 5² = -7

For you to work this problem as:

4 x 2³ + -5² = 4 x 2³ + 25 = 54 !WRONG!

You have to break the basic rules of math by changing the order of operation or fill in an assumed set of parenthesis that just are present in the problem. Therefore, it should be apparenent that the correct answers to the problems are -7 and 18!

OH SNAP! He broke out the commutative property on your ###!

I LOVE this thread.

Has there ever been a 16 page argument where someone wasn't called a racist before?

 

[MrPack]

Wow, 15 pages when the correct answer was provided with the first reply.

It's an extremely interesting discussion.

 

[Clayton Gray]

-5² + 4 x 2³ = 4 x 2³ + -5²

Now it should be obvious that:

4 x 2³ + -5² = 4 x 2³ - 5² = -7

No no, you've disobeyed the order of operations laws here.You can't convert "... + -5^2" to "... - 5^2" until after you resolve the exponent.

IncorrectInexact

Inexacto

Falsch

Inesatto

Verkeerd

Incorreto

Неправильный

Ukorrekt

不正确

Only according to your arbitrary convention. According to intuition and logic, I am right.

The only thing that matters is that there is a proper way to compute -5^2, and I am advocating that way.

 

[shuke]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

But he's proving that there's no way to know if it is part of the number or an operator UNLESS THERE ARE PARENTHESES.

 

[Gamblor]

This alludes back to an earlier post, but what would your answers be for these?

4 x 2³ - 5² =

2 x 3² - 6² =

4*8-25= 7But, had it been presented as -5^2 + 4*2^3 I would've said 25+32=57

But the two should be equal:4 x 2³ - 5² = -5² + 4 x 2³

Left side is 7. Right side is 57.NO, the 2 are not equal.

As mentioned in this earlier post by Iguana, they should be:

The answer lies in the commutative property of addition.

Given that a + b = b + a:

-5² + 4 x 2³ = 4 x 2³ + -5²

Now it should be obvious that:

4 x 2³ + -5² = 4 x 2³ - 5² = -7

For you to work this problem as:

4 x 2³ + -5² = 4 x 2³ + 25 = 54 !WRONG!

You have to break the basic rules of math by changing the order of operation or fill in an assumed set of parenthesis that just are present in the problem. Therefore, it should be apparenent that the correct answers to the problems are -7 and 18!

OH SNAP! He broke out the commutative property on your ###!

I LOVE this thread.

Has there ever been a 16 page argument where someone wasn't called a racist before?

 

[Clayton Gray]

- 5^2 <> -5^2

I don't think spaces have any meaning in math...

Then why use them? They were inserted into the last problem for a reason: to separate operators from numbers. So why did the -5^2 not receive the same space? Because the - isn't an operator in that case.

A negative sign is pretty much the same thing as a minus sign.In -5^2, what is in front of the negative/minus sign? Nothing.

What is the number for nothing? Zero.

So -5^2 = 0-5^2.

This is true if you view the minus sign as an operator. It's not, in this case. It's part of the integer '-5'.

Actually, it's true if you view the minus sign correctly.

 

[shuke]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

-5^2 = x-therefore-

0 = x - (-5^2)

x = 25

 

[Smoo]

- 5^2 <> -5^2

I don't think spaces have any meaning in math...

Then why use them? They were inserted into the last problem for a reason: to separate operators from numbers. So why did the -5^2 not receive the same space? Because the - isn't an operator in that case.

A negative sign is pretty much the same thing as a minus sign.In -5^2, what is in front of the negative/minus sign? Nothing.

What is the number for nothing? Zero.

So -5^2 = 0-5^2.

This is true if you view the minus sign as an operator. It's not, in this case. It's part of the integer '-5'.

Actually, it's true if you view the minus sign correctly.

If by 'correctly', you mean 'non-intuitive, incorrect, illogical nerd consensus' then I agree.

 

[Clayton Gray]

You can't convert "... + -5^2" to "... - 5^2" until after you resolve the exponent.

You are completely wrong here.

I'm totally right, because they're different problems.... + -5^2 = ... + 25

... - 5^2 = ... - 25

Very different.

-5^2 = (-1)*(5^2), and I don't care if you put a + sign in front of it or not.

Nope. Exponents come before multiplication in the order of operations. You cannot factor out the -1 (an act of multiplication/division) until after resolving the exponent.

Nope. Exponents come before multiplication in the order of operations. You cannot multiply the -1 until after resolving the exponent.

 

[Smoo]

Holy cow, this thread has legs. Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

-5^2 = x-therefore-

0 = x - (-5^2)

x = 25

You have a counterpoint?

 

[Joseph]

When I read it, I saw it as negative 5... squared. I graduated in 99 with a degree in mathematics, and I didn't even consider that the negative would be an operator in that context. If I wanted to convey that it was negative ... 5 squared I would have written -(5^2) or 0 - 5^2 maybe they do things differently now though.

 

[Ignoramus]

Holy cow, this thread has legs. Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

For what? This is basic arithmetic; there is no need to take a vote on it or discuss.

 

[Smoo]

You can't convert "... + -5^2" to "... - 5^2" until after you resolve the exponent.

You are completely wrong here.

I'm totally right, because they're different problems.... + -5^2 = ... + 25

... - 5^2 = ... - 25

Very different.

-5^2 = (-1)*(5^2), and I don't care if you put a + sign in front of it or not.

Nope. Exponents come before multiplication in the order of operations. You cannot factor out the -1 (an act of multiplication/division) until after resolving the exponent.

Nope. Exponents come before multiplication in the order of operations. You cannot multiply the -1 until after resolving the exponent.

There is no -1 until after doing a division operation. There's only a -5. You can't do the division operation to extract the -1 until after the exponent is resolved.

 

[Clayton Gray]

- 5^2 <> -5^2

I don't think spaces have any meaning in math...

Then why use them? They were inserted into the last problem for a reason: to separate operators from numbers. So why did the -5^2 not receive the same space? Because the - isn't an operator in that case.

A negative sign is pretty much the same thing as a minus sign.In -5^2, what is in front of the negative/minus sign? Nothing.

What is the number for nothing? Zero.

So -5^2 = 0-5^2.

This is true if you view the minus sign as an operator. It's not, in this case. It's part of the integer '-5'.

Actually, it's true if you view the minus sign correctly.

If by 'correctly', you mean 'non-intuitive, incorrect, illogical nerd consensus' then I agree.

No. In this case, I actually meant 'correctly'.

 

[shuke]

Look people, it's this simple.If you are performing operations on a negative number, parentheses are necessary for clarification that the negative sign is part of the integer and not an operation. If you don't put the parentheses there it's assumed it's an operation.I don't get this "new math" talk. This is how I was taught 25 years ago.Quit being a bunch of morons.

 

[chet]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

For what? This is basic arithmetic; there is no need to take a vote on it or discuss.

Ahh, I see how you got your name.

 

[Clayton Gray]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

-5^2 = x-therefore-

0 = x - (-5^2)

x = 25

You have a counterpoint?

I have one: X

 

[Maurile Tremblay]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2). Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

I figured we were all on the same page once the math teachers got here.

 

[Smoo]

If you don't put the parentheses there it's assumed it's an operation.

Don't be a bigot. Anti-negativist.

 

[Clayton Gray]

I would like to point out that this is a middle school concept. People that were math nerds in middle school often bypassed the regular math classes and went straight to Algebra. It is completely possible that they missed being taught this and are relying upon their incorrect intuition.

 

[NCCommish]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2).  Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

I figured we were all on the same page once the math teachers got here.

Experts? We don't need no stinking experts!

 

[MrPack]

When I read it, I saw it as negative 5... squared. I graduated in 99 with a degree in mathematics, and I didn't even consider that the negative would be an operator in that context. If I wanted to convey that it was negative ... 5 squared I would have written -(5^2) or 0 - 5^2

maybe they do things differently now though.

Evidentally this is true, because I saw it the same way you did.

 

[Smoo]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2). Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

I figured we were all on the same page once the math teachers got here.

I'm role-playing one of the people at the table during the conference where this arbitrary decision was made. We've established what the convention is. I'm trying to get them to reason it out. My goal here is to make it clear that the "convention" was decided poorly.

 

[Clayton Gray]

You can't convert "... + -5^2" to "... - 5^2" until after you resolve the exponent.

You are completely wrong here.

I'm totally right, because they're different problems.... + -5^2 = ... + 25

... - 5^2 = ... - 25

Very different.

-5^2 = (-1)*(5^2), and I don't care if you put a + sign in front of it or not.

Nope. Exponents come before multiplication in the order of operations. You cannot factor out the -1 (an act of multiplication/division) until after resolving the exponent.

Nope. Exponents come before multiplication in the order of operations. You cannot multiply the -1 until after resolving the exponent.

There is no -1 until after doing a division operation.

Yes there is.

 

[Gamblor]

Holy cow, this thread has legs. Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

-5^2 = x-therefore-

0 = x - (-5^2)

x = 25

You have a counterpoint?

I have one: X

 

[MrPack]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

For what? This is basic arithmetic; there is no need to take a vote on it or discuss.

Ahh, I see how you got your name.

Careful there chetski, I get a 48hr timeout for that 2 weeks ago.

 

[Clayton Gray]

Look people, it's this simple.

If you are performing operations on a negative number, parentheses are necessary for clarification that the negative sign is part of the integer and not an operation. If you don't put the parentheses there it's assumed it's an operation.

I don't get this "new math" talk. This is how I was taught 25 years ago.

Quit being a bunch of morons.

As usual, Shuke nails it.

 

[shadyridr]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

-5^2 = x-therefore-

0 = x - (-5^2)

x = 25

ok If we did math like this then all of our bridges would be collapsed.

 

[Maurile Tremblay]

I disagree. The problem as written has a very clear and definite meaning. There is no ambiguity here.

The negative sign out front is ambiguous - especially to a child learning operations.

What's funny to me is that we've got a bunch of people who agree that the negative sign is not ambiguous -- but half of these people think it unambiguously means (-5)^2 and the other half think it unambiguously means -(5^2).It seems to be ambiguous.

 

[Ignoramus]

Holy cow, this thread has legs. Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

For what? This is basic arithmetic; there is no need to take a vote on it or discuss.

Ahh, I see how you got your name.

Good one. I'm going to write that down and use it sometime.

 

[Clayton Gray]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

For what? This is basic arithmetic; there is no need to take a vote on it or discuss.

Ahh, I see how you got your name.

But people that know the answer think -5^2 = 25 is as correct as saying a chair is a couch. One might make that mistake before being taught the difference.

 

[chet]

Real world application.

 

[(HULK)]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

But he's proving that there's no way to know if it is part of the number or an operator UNLESS THERE ARE PARENTHESES.

Really.Please read this number out loud: -5

What did you just say?

 

[shadyridr]

- 5^2 <> -5^2

I don't think spaces have any meaning in math...

Then why use them? They were inserted into the last problem for a reason: to separate operators from numbers. So why did the -5^2 not receive the same space? Because the - isn't an operator in that case.

A negative sign is pretty much the same thing as a minus sign.In -5^2, what is in front of the negative/minus sign? Nothing.

What is the number for nothing? Zero.

So -5^2 = 0-5^2.

This is true if you view the minus sign as an operator. It's not, in this case. It's part of the integer '-5'.

It IS an operator

 

[Maurile Tremblay]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2). Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

No the base number is 5. If the base number was -5 THEN you would use parentheses around the -5

No, the base number is -5. Self-contained numbers don't require parentheses.All this thread has established is that the powers-that-be in math have established by consensus that negative numbers are not to be considered actual numbers, merely as positive numbers with an operation on them. That doesn't make them right, it just makes them bullies with a stupid idea.

But if it's a consensus, it's right. That's how arbitrary conventions work.You may as well argue that we should do addition before multiplication. The only reason we don't is that the powers-that-be in math have established by consensus that multiplication comes before addition.

 

[shadyridr]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

But he's proving that there's no way to know if it is part of the number or an operator UNLESS THERE ARE PARENTHESES.

Really.Please read this number out loud: -5

What did you just say?

Who cares how the #### you say it? That's just for proper sounding discussions. Its how its done on a piece of paper that makes it accurate.

 

[Gamblor]

Really.

Please read this number out loud: -5

What did you just say?

5 times negative 1

 

[P Boy]

When I read it, I saw it as negative 5... squared. I graduated in 99 with a degree in mathematics, and I didn't even consider that the negative would be an operator in that context. If I wanted to convey that it was negative ... 5 squared I would have written -(5^2) or 0 - 5^2

maybe they do things differently now though.

Maybe smoo will hire you to be his accountant. I'll bet you'll make his financial situation look better in no time at all!

 

[Clayton Gray]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2).  Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

I figured we were all on the same page once the math teachers got here.

I'm role-playing one of the people at the table during the conference where this arbitrary decision was made. We've established what the convention is. I'm trying to get them to reason it out. My goal here is to make it clear that the "convention" was decided poorly.

Oh.We're trying to simply things. Using -(5)^2 is simply too messy when -5^2 will suffice. If we want to square the negative, we'll write (-5)^2. So let it be written, so let it be convention.

Now, let's decide which letter will be most used to depict a variable.

 

[Smoo]

But if it's a consensus, it's right.

Oh please. You should know as well as anybody that the tyranny of democracy doesn't equate to 'right'.

 

[Smoo]

You can't convert "... + -5^2" to "... - 5^2" until after you resolve the exponent.

You are completely wrong here.

I'm totally right, because they're different problems.... + -5^2 = ... + 25

... - 5^2 = ... - 25

Very different.

-5^2 = (-1)*(5^2), and I don't care if you put a + sign in front of it or not.

Nope. Exponents come before multiplication in the order of operations. You cannot factor out the -1 (an act of multiplication/division) until after resolving the exponent.

Nope. Exponents come before multiplication in the order of operations. You cannot multiply the -1 until after resolving the exponent.

There is no -1 until after doing a division operation.

Yes there is.

I don't see it. I see a -5. Every example I've seen that introduces a -1 does so by a factoring operation, which falls in the order of operations under "multiplication/division", decidedly after exponents.

 

[Maurile Tremblay]

If the problem was meant to be read the negative of 5 squared, then it necessitated a parentheses. Bottom line.

This is the answer to this whole entire thread.Parentheses would tell you one thing. The LACK of parentheses tells you another.

It's clear-cut.

I was leaning the other way last night, but after reading what smoo smoo wrote and now what hulk wrote, I am moving this way.Why isn't the -5 the base number?

Because "-" isn't a number. "5" is a number, and "-" is an operator.(And before Smoo attacks me, I'm not addressing this to Smoo. Smoo already understands the operator argument. I'm addressing this to the many people who still don't understand the argument.)

 

[Ignoramus]

Holy cow, this thread has legs. Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

But he's proving that there's no way to know if it is part of the number or an operator UNLESS THERE ARE PARENTHESES.

Really.Please read this number out loud: -5

What did you just say?

This isn't reading, it's arithmetic. The beauty is in the proof.-5^2=x

Prove to me that x=25

 

[Smoo]

To force the processor to intrepret it correctly would mean changing every formula that includes exponentiation to check for a negative base number and if found change the formula from something like x^2 to (-1*(x)^2). Otherwise, it comes out positive every time.

No, a negative base number should come out positive when it is squared. But in the expression -5^2, the base number (i.e., the number that gets squared) isn't negative. It's five.

Wrong. And your continuing to assert this as divine truth is getting mildly irritating. The base number in this problem is -5. -5 is a number, all on its own. If you want to spedify that it's an operation, you need to do so explicitly, using parentheses.-5^2 = 25 exactly follows the order of operations, because there's only a SINGLE OPERATION.

I figured we were all on the same page once the math teachers got here.

I'm role-playing one of the people at the table during the conference where this arbitrary decision was made. We've established what the convention is. I'm trying to get them to reason it out. My goal here is to make it clear that the "convention" was decided poorly.

Oh.We're trying to simply things. Using -(5)^2 is simply too messy when -5^2 will suffice. If we want to square the negative, we'll write (-5)^2. So let it be written, so let it be convention.

Now, let's decide which letter will be most used to depict a variable.

This only works if you presuppose that numbers cannot be inherently negative. What's your reasoning for that?

 

[(HULK)]

Really.

Please read this number out loud:  -5

What did you just say?

5 times negative 1

Well played sir, well played.

I just don't agree with the logic that in one instance its read as negative five, and in another instance its read as negative one times five. That is poor logic.

 

[Clayton Gray]

Oot for a while.

 

[bryhamm]

Holy cow, this thread has legs.   Just goes to show how lousy most people are at math, even smart people.

-5^2=x

-therefore-

0=x+5^2

Anyone here still think x=25?

Please see discussion on whether the - is part of the number or an operator. TIA.

But he's proving that there's no way to know if it is part of the number or an operator UNLESS THERE ARE PARENTHESES.

Really.Please read this number out loud: -5

What did you just say?

"minus 5", why?

 

[Smoo]

If the problem was meant to be read the negative of 5 squared, then it necessitated a parentheses. Bottom line.

This is the answer to this whole entire thread.Parentheses would tell you one thing. The LACK of parentheses tells you another.

It's clear-cut.

I was leaning the other way last night, but after reading what smoo smoo wrote and now what hulk wrote, I am moving this way.Why isn't the -5 the base number?

Because "-" isn't a number. "5" is a number, and "-" is an operator.(And before Smoo attacks me, I'm not addressing this to Smoo. Smoo already understands the operator argument. I'm addressing this to the many people who still don't understand the argument.)

I'd have thought you'd have wanted to see another the-people-will-rise, stop-oppressing-the-negatives call to arms post. You seem to enjoy them.

 

[Bottomfeeder Sports]

I would like to point out that this is a middle school concept.  People that were math nerds in middle school often bypassed the regular math classes and went straight to Algebra.  It is completely possible that they missed being taught this and are relying upon their incorrect intuition.

So - if you get -25 you were either not very good at math so you went to the basics classes in 8th grade on, or you are trained as a math teachers. (What is that cliche about those who teach?) But you get feel all superior about technically being correct.If you get 25 you skipped over learning a silly rule which over complicates what should be a simple expression, because you were capable of understanding math.

By the way -5^2 is not math at all, it is arithmetic!

 

[Maurile Tremblay]

From the Prentice Hall "Middle Grades Math - Tools for Success" Course 2 - 1999 Edition, Page 157:

When you use an exponent with a negative numer as the base, it is important to use grouping symbols to avoid confusion.

Example 3: Simply each expression.

a. (-5)^4 = (-5)(-5)(-5)(-5) = 625

b. -5^4 = -(5*5*5*5) = -625

This should definitively establish that -5^2 doesn't unambiguously mean (-5)^2.