How to calculate negative exponents?

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mathnoob2022

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I thought that -2^2 = 4. Because negative number times negative number equals positive number: -2 x -2 = 4.

Let's take another example: -3^4 = -81 and not 81. I don't understand the logic here.
 
mathnoob2022 said:
I thought that -2^2 = 4. Because negative number times negative number equals positive number: -2 x -2 = 4.

Let's take another example: -3^4 = -81 and not 81. I don't understand the logic here.
Click to expand...
Yes ... but -2^2 is assumed to be -(2^2) which is equal to -4.

If you wanted the output to be 4, you need to write it as (-2)^2 → (-2)* (-2) = 4

This all done as convention or definition (just like √4 = 2 ..... it is taken to be positive while we use "radical" sign.

similarly -3^4 → -(3^4) → -81 (negative 81)

and (-3)^4 → (-3)*(-3)*(-3)*(-3) = 81 (positive 81)
 
mathnoob2022 said:
I thought that -2^2 = 4. Because negative number times negative number equals positive number: -2 x -2 = 4.

Let's take another example: -3^4 = -81 and not 81. I don't understand the logic here.
Click to expand...
To reinforce Mr. Khan's point: Order of operations. PEDMAS (or BODMAS.) Exponentiation occurs before addition, so the negative sign out front happens after we square. So −22=(−)22=(−)2⋅2=−4-2^2 = (-)2^2 = (-) 2 \cdot 2 = - 4 whereas (−2)2=(−2)⋅(−2)=4(-2)^2 = (-2) \cdot (-2) = 4

-Dan
 
mathnoob2022 said:
I thought that -2^2 = 4. Because negative number times negative number equals positive number: -2 x -2 = 4.

Let's take another example: -3^4 = -81 and not 81. I don't understand the logic here.
Click to expand...
The idea behind the convention that -2^2 means -(2^2) is that negation is treated like subtraction, which is done after exponents and multiplication. In particular, if you write -x^2, you square x and then take the negative.

So this is all about how we interpret the written expression. No negative number is being squared unless you make it explicit by writing (-2)^2.
 
What if you had 33-4^2? Would you say that it equals 33+16=49? Probably not! You would probably say 33-4^2= 33-16 = 17. That is because -4^2 = -16.
I like to think of -4^2 as 0-4^2 = 0 -16 = -16.

We all know that 4^2 = 16 and if we put a negative sign in front of 16 we get -16.
So -4^2 = -16
 
BTW, none of your examples had a negative exponent. 3-2 has a negative exponent.
 
I agree with the posts above. However, some computer applications - not many - are programmed to interpret as (-2)^2. In particular, most spreadsheets will actually interpret this way :eek: .

Try typing these into a spreadsheet:- =-2^2 and =0-2^2. The first gives 4 and the second -4. If in doubt, use brackets.
 
I am not sure that this is helpful to students when all is considered, but I always tell the kids whom I tutor in person that math notation is a language and, like all languages, most of the rules of a language’s grammar are arbitrary. Order of operations are not inherent in the universe; they are a way of talking to each other without confusion and must be memorized like sum, es, est, sumus, estis, sunt or am, are, is, are, are, are. The rule is:

−a2<0<(−a)2=a2  ⟺  a≠0.- a^2 < 0 < (-a)^2 = a^2 \iff a \ne 0.
Memorize it.

Students want to know why? There may be conveniences in PEMDAS, but there is no logical necessity: “a rose by any other name would smell as sweet,” Why is it bad English grammar to say “she am”? Because it is.
 
I think that it is natural the way this plays out. 16 is 16. Now if we put a negative sign in front of it, it becomes -16.
Now 4^2 is 16. We all know that. Now if we put a negative sign in front of it, it becomes -16. That is, -4^2 = -16

Again, as I said earlier in this post, 33-4^2 = 33-16=17. Most students will get this correct. When the 33 is removed is when the problem starts. Students just need to realize that 33-4^2 = 33-16 because -4^2 = -16. OR if the negative sign is not inside the bracket with the number like in (-3)^2=9
 
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