Which is larger?

By the way, is there any fast way to do this?

Lets see

$\frac{53^{2000}}{53^{1999}}=53^{2000-1999}=53$

Think about the second one.

The numerator and the denominator are reduced by 2000

    but proportionately the numerator has been reduced by less than the denominator because it was bigger to start with.

   so proportionately... the top was reduced by a smaller factor than the bottom so the entire number got bigger.

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It is like this

$\frac{10}{8}\;\;compared \;\;to \;\;\frac{9}{7}\\ 1\frac{2}{8}\;\;compared \;\;to \;\;1\frac{2}{7}\\ $

The second one is bigger.

ohhhh i understand thank you