# What practical reason could there be for teaching the difference parenthesis make with regard to negative number bases with exponents?

This question came up at work yesterday, and my co-worker and I kind of wrestled with it. Basically, everything that I’m seeing says that when dealing with a negative base, when it has parenthesis around it, if the exponent is an even number, the answer will be a positive number. If the exponent is odd, the answer will be a negative number. So (-3)^2 = 9 ........... (-3)^3 = -27 (-3)^4 = 81. Pretty straight forward. Until you take the parenthesis away. From what I’m seeing, if there are no parenthesis the answer will ALWAYS be negative. So….... -3^2 = -9. =3^4 = -81

See Purple Math

That is easy enough to remember I suppose..but WHY? I don’t recall ever learning this in math classes I’ve taken in the past. I noticed in Purple Math, at the bottom of the paragraph of explanations it specifically mentions programming. Is this type of subtlety programming specific? If so, why should we even teach it? Is there a common application where this difference may be used?

It seems to me that it creates a tempest in a tea pot. Confuses things that are already confusing for a student.

**I’m asking your opinions so I can determine the best way to teach this to my students.** If I can mention the difference to them, then say, “But for our purposes lets always assume an odd exponent yields a negative answer, and an even exponent yields a positive answer, etc….” I’d rather be able to do that.

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## 11 Answers

Isn’t this common knowledge that fits in nicely with order of operations? It is what I was always taught.

What do you mean? I know the order of operations. I don’t understand how it applies in this case.

HELLLLLOOOO!!!!! Anybody out there? Ivan? Joe? Somebody?

Guess it isn’t common knowledge, @johnpowell!

Operator precedence is important in any setting where mathematics is used. Doing arithmetic is actually the application of operators to one or more numbers yielding another number, and the order in which the operations are carried out determines the result. Precedence is just a set of conventions to simplify the process of deciding what to first, second, and so on. Consider:

5 + 8 * 3 – 12 .

The operators are addition, multiplication, and subtraction. Which should be applied first? does it make a difference?

(((5 + 8 ) * 3) – 12) = ((13 * 3) – 12) = (39 – 12) = 27.

(((5 + (8 * 3)) -12) = ((5 + 24) – 12) = (29 – 12) = 17.

((5 + 8) * (3 – 12)) = (13 * (3 – 12)) = ((13 * 3) – (13 * 12)) = 29 – 156 = – 127.

In the case of exponents, it’s helpful to think of negation as subtraction; i.e., – 3 = (0 – 3), and recall that – - 3 = 3. Thus:

(- 3) ^ 2 = (0 – 3) * (0 – 3) = ((0 – 3) * 0 – (0 – 3) * 3) = (0 – (0 – 3) * 3) =

(0 – ((0 * 3) – (3 * 3)) = (0 – (0 – 9)) = ( – ( 0 – 9)) = – – 9 = 9, while

- (3 ^ 2) = (0 – (3 * 3)) = (0 – 9) = – 9.

Summary: When the negative sign is outside the parentheses, exponentiate first and negate second. When the negative sign is inside the parentheses, negate first and exponentiate second.

I understand the order of operations, @ratboy. I understand how to calculate negative bases with exponents both ways, and I understand why. My question is, is there any reason to teach both ways, if one way is mostly used in some obscure purpose, like computer programming? I’m not into confusing my students, like it seems so many text books are.

There are not two ways of calculating powers involving negative numbers. Perhaps what hasn’t been stressed is that parentheses override the default precedence of operations.By default, exponentiation is done before multiplication and division. Multiplication and division are on the same level. Multiplication and division are done before addition and subtraction. Addition and subtraction are on the same level. As above, negation is the same as subtraction and is done after exponentiation, multiplication, and division. This is the standard convention, although not everyone adheres to it.

Thus -3^2 = -9 because, by convention, exponentiation is done before negation, and (-3)^2 = 9 because the parentheses indicate a departure from convention in that negation is to precede exponentiation. This is basic arithmetic; the concepts are neither difficult nor confusing.

Nicely explained @ratboy. That’s simple enough to explain to my students (without using words like ‘negation’ and ‘precede’ and ‘convention.’ IMO, those types of words, instead of “negative,” ‘before,” and “change from the usual way,” confuse things more. I mean, you have a student who isn’t at the highest reading level in the world, and he’s now struggling with more complex math AND reading comprehension at the same time. That is a big problem with some textbooks.) Anyway, thanks.

It’s called “PEDMAS”. Parentheses first. If no parentheses then the exponent is next. Add/Subtract comes last.

I know what it is @SmartAZ. I’m a teacher. That wasn’t the question.

I’m a teacher too. You need to work on presentation.

I should have used smaller words. I’m sorry if I confused you.

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