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Thread: Negative exponent flipping question

  1. #1
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    Negative exponent flipping question

    Hello all!

    I'm preparing for an Algebra test on Monday. I understand that $\displaystyle (2)^{-3} = \frac {1}{2^3}$ ultimately resulting in $\displaystyle \frac{1}{8}$ and I understand "why." Our teacher gave me a really good explanation starting with $\displaystyle (2)^{0}=1$ and worked down from there, I finally got it!


    But what I don't seem to understand is why, $\displaystyle \Big( \frac{2}{z}\Big)^{-3}=\frac{z^3}{8}$. If I had to break it down I would start with, $\displaystyle \Big( \frac{2}{z}\Big)^{-3} = \Big( \frac{{2^{-3}}}{{z^{-3}}} \Big)$ but I get lost after that because I don't really understand why the exponents subtract from each other (or add) and why.

    Do the -3 cancel each other out? (e.g, -3(+3)) that would result in 0.

    So what exactly happens when they flip?

    Thank you
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  2. #2
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    Re: Negative exponent flipping question

    (2/x)^(-3) = 1 / (2/z)^3 = 1 / (8/z^3) ..... HOKAY?
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  3. #3
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    Re: Negative exponent flipping question

    Quote Originally Posted by alexcordero View Post
    Hello all!

    I'm preparing for an Algebra test on Monday. I understand that $\displaystyle (2)^{-3} = \frac {1}{2^3}$ ultimately resulting in $\displaystyle \frac{1}{8}$ and I understand "why." Our teacher gave me a really good explanation starting with $\displaystyle (2)^{0}=1$ and worked down from there, I finally got it!


    But what I don't seem to understand is why, $\displaystyle \Big( \frac{2}{z}\Big)^{-3}=\frac{z^3}{8}$. If I had to break it down I would start with, $\displaystyle \Big( \frac{2}{z}\Big)^{-3} = \Big( \frac{{2^{-3}}}{{z^{-3}}} \Big)$
    Good! So you have $\displaystyle \frac{\frac{1}{8}}{\frac{1}{z^3}}$. Now some time ago you surely learned that "to divide fractions, invert the divisor and multiply". So $\displaystyle \frac{\frac{1}{8}}{\frac{1}{z^3}}= \frac{1}{8}\frac{z^3}{1}= \frac{z^3}{8}$.

    but I get lost after that because I don't really understand why the exponents subtract from each other (or add) and why.

    Do the -3 cancel each other out? (e.g, -3(+3)) that would result in 0.
    $\displaystyle (a^x)(a^y)= a^{x+y}$ only for the same base, a. Here you have $\displaystyle z^3$ and $\displaystyle 2^3$. The bases are different so you cannot combine exponents.

    So what exactly happens when they flip?

    Thank you
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