Removing the exponent

Posted on 2011-04-18
Last Modified: 2012-05-11
This is the question and the answer, but I don't understand it.

With an exponent of ^2 it's a simple matter -3(-2)^2 = -3(4). But the problem below has both a negative exponent and an exponent less than 2. Can someone help me understand first how a negative exponent effects a problem and then how an exponent of less than 2 effects a problem? Thanks.

-3(-2)^-1 = -(3/-2)
Question by:kadin
    LVL 31

    Assisted Solution

    a^(-1) is defined to be 1/a

    -3 * (a)^(-1) = -3/a

    Let a = -2
    -3 * (-2)^(-1) = -3/(-2) = +3/2
    LVL 31

    Accepted Solution

    Now consider a^(-2). That is just 1/( a^2 )
    So, if a = -2, then a^(-2) = 1/( a^2 ) = 1/(   (-2)^2   ) = 1/(   -2*-2 ) = +1/4
    LVL 31

    Assisted Solution

    Here is something that may help you understand why the negative exponent was defined that way.

    Consider a^5/a^3 = (aaaaa)/aaa = aa = a^2
    which leads to a natural formula to remember:  a^n / a^k = a^(n-k)i.e., a^5/a^3 = a^(5-3) = a^2

    Now consider a^3/a^5 = aaa/(aaaaa) = 1/(aa) = 1/a^2
    But, if you try to use the above formula, you get a^3/a^5 = a^(3-5) = a^(-2)

    So, someone said, let's define a negative exponent to be:  a^(-n) = 1/a^n
    and since this one formula seemed to fit well into algebra, it stuck.
    LVL 31

    Expert Comment

    Here's a little more discussion of negative exponents:

    Author Comment

    Thanks for your help. I think I understand this a little better.
    LVL 31

    Expert Comment

    You're welcome.
    BTW - I tend not to try to remember a lot of formulas. Instead I try to understand their derivation, and use that approach when thinking about these types of problems.

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