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

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

    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 by Dinty_; 11-14-2017 at 07:38 PM.

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    Quote Originally Posted by Dinty_ View Post
    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
    ... mathematics is only the art of saying the same thing in different words - B. Russell

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    Quote Originally Posted by Dinty_ View Post
    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.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

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    Quote Originally Posted by Dinty_ View Post
    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.)

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    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by Dinty_ View Post
    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.

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    Quote Originally Posted by stapel View Post
    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 ?

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    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by Dinty_ View Post
    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:

    (ab)^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.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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