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$\left(\dfrac 25\right)^{-3} = \dfrac{1}{\left(\frac 25\right)^3} = \dfrac{1}{\frac 8{125}} = \dfrac {125}8$

I'm not sure if my intuition about the above is correct, but here goes:
My intuition tells me that since we are taking the reciprocal of a fraction, the numerator of the "higher fraction" ($1$) is the whole of the denominator of the "lower fraction" ($125$), and so we can move the denominator of the lower fraction as the numerator of the higher fraction, because $1$ is practically the same as $125$.

What I can't seem to grasp is then why does the numerator of the lower fraction ($8$) become the denominator of the new fraction?

My question is very much like this one, but rotated.

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  • $\begingroup$ Multiply the second last fraction top and bottom with $\frac{125}{8}$ $\endgroup$ – imranfat Feb 28 '18 at 22:06
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So you don't understand this part:

$$\dfrac{1}{\frac 8{125}}=1 \div\dfrac 8{125}=1\times \dfrac{125}8=\dfrac {125}{8}$$

Can you understand it from there?

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    $\begingroup$ Just reformulating the fraction doesn't really explain the intution (or how you should be thinking about it). I know how to plot the above into a calculator, doesn't mean that I understand what i'm plotting in. $\endgroup$ – user3407764 Feb 28 '18 at 22:09
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By definition $\frac{1}{a}$ is the inverse of $a$, i.e. the number such that $a \cdot \frac{1}{a}=1$.

So, what is the inverse of $\frac{8}{125}$?

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  • $\begingroup$ The inverse is 1/8/125, yes? So, how does it explain the flip? $\endgroup$ – user3407764 Feb 28 '18 at 23:17
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I managed to make sense of this one by looking at the fractions from a decimal point of view.

$\dfrac {1}{\frac {8}{125}} = \dfrac {1}{0.064}$

$1$ is $15.625$ times larger than $0.064$. So that also means that $125$ is $15.625$ times larger than $8$. Thus,

$\dfrac {1}{\frac {8}{125}} = \dfrac {1}{0.064} = \dfrac {125}{8}$

While I appreciate the other guys' answers, (and some will probably feel they explained it sufficiently) I found the above to give the "intuitive leap" to better understanding why the bottom fraction is flipped up, for someone not well versed in mathematics.

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  • $\begingroup$ This is just the same with the other answers, just that it was transformed into decimals $\endgroup$ – John Glenn Mar 1 '18 at 1:52

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