Why does the Numerator of the Denominator go to the Denominator? (2/5)^(-3) =1/(2/5)^3 = 1/(8/(125)) = (125)/8

Why does the Numerator of the Denominator go to the Denominator?
${\left(\frac{2}{5}\right)}^{-3}=\frac{1}{{\left(\frac{2}{5}\right)}^{3}}=\frac{1}{\frac{8}{125}}=\frac{125}{8}$
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?
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Isaac Molina
So you don't understand this part:
$\frac{1}{\frac{8}{125}}=1÷\frac{8}{125}=1×\frac{125}{8}=\frac{125}{8}$
Can you understand it from there?
Did you like this example?
unjulpild9b
I managed to make sense of this one by looking at the fractions from a decimal point of view.
$\frac{1}{\frac{8}{125}}=\frac{1}{0.064}$
1 is 15.625 times larger than 0.064. So that also means that 125 is 15.625 times larger than 8. Thus,
$\frac{1}{\frac{8}{125}}=\frac{1}{0.064}=\frac{125}{8}$
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.