# Question about exponents: why are -2^4, (-2)^4 diff, but -2^3, (-2)^3 are same?

#### Dinty_

##### New member
Hey guys,

Why is it that the answer to -2^4 and (-2)^4 is different, but the answer to -2^3 and (-2)^3 is the same?

At first, I thought it had something to do with one side being raised to an even number and the other to an odd number.

If that were the case, both sides should have the same answer.

So, I went riffling through my notes and remembered that if the sign is outside parenthesis, it is not considered part of the base.

So I tried it out:

-2^4 = -1 x -2 x -2 x -2 x -2 = -16

(-2)^4 = (-2)(-2)(-2)(-2) = 16

But if I apply the same rule to the other side, it gives me two different answers:

-2^3 = -1 x -2 x -2 x -2 = 8

(-2)^3 =
(-2)(-2)(-2) = -8

I am so confused! am I doing something wrong here?

Last edited:

#### Subhotosh Khan

##### Super Moderator
Staff member
Hey guys,

Why is it that the answer to -2^4 and (-2)^4 is different, but the answer to -2^3 and (-2)^3 is the same?
What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

#### tkhunny

##### Moderator
Staff member
Hey guys,

Why is it that the answer to -2^4 and (-2)^4 is different, but the answer to -2^3 and (-2)^3 is the same?
This depends on your environment. Look up the "Precedence" of "Unary Minus".

-2^4 can mean either (-2)^4 or -(2^4), depending on how the sequence of charters is interpreted. You have to know where you are writing.

#### Dr.Peterson

##### Elite Member
Hey guys,

Why is it that the answer to -2^4 and (-2)^4 is different, but the answer to -2^3 and (-2)^3 is the same?
Presumably you are following the common convention that negation is done after exponents, so that what you are asking about is true.

Show us how you evaluate each of the four expressions, and we can discuss what it is that makes the difference. (Hint: it has to do with odd and even.)

#### stapel

##### Super Moderator
Staff member
Why is it that the answer to -2^4 and (-2)^4 is different, but the answer to -2^3 and (-2)^3 is the same?
What do you know about negative numbers, and even (like 2, 4, 6) versus odd (like 3, 5, 7) powers? What do even powers do to negative numbers, than odd powers don't?

Then think about grouping symbols, and what they tell out about what is included in the "base" on which the power has been put. #### Dinty_

##### New member
What do you know about negative numbers, and even (like 2, 4, 6) versus odd (like 3, 5, 7) powers? What do even powers do to negative numbers, than odd powers don't?

Then think about grouping symbols, and what they tell out about what is included in the "base" on which the power has been put. As long as the base is in parentheses, a negative number paired with an even exponent will result in a positive answer, but a negative number paired with an odd exponent will result in a negative answer ?

#### mmm4444bot

##### Super Moderator
Staff member
As long as the base is in parentheses, a negative [base raised to] an even exponent will result in a positive answer, but a negative [base raised to] an odd exponent will result in a negative answer ?
Yes. The factorizations below are related to what DrPeterson asked you about.

First, know that the negative sign in front of a number can be viewed as a factor of -1. For example, -2 can be thought of as (-1)(2).

There's also a property of exponents for factored bases:

(a·b)^n = a^n · b^n

So, we can write:

-2^4 = (-1)(2)(2)(2)(2) = -16

(-2)^4 = (-1)(-1)(-1)(-1)(2)(2)(2)(2) = 16

-2^3 = (-1)(2)(2)(2) = -8

(-2)^3 = (-1)(-1)(-1)(2)(2)(2) = -8

Think about those factors.