Finding the exact value of each number goes like this:
x = 0.212121...
100x = 21.212121...
==> 100x - x = 21.212121... - 0.212121...
==> 99x = 21
==> x = 21/99 = 7/33
Similarly:
y = 0.021021021...
1000y = 21.021021021...
==> 999y = 21
==> y = 21/999 = 7/333
and
z = 0.145714571457...
10000z = 1457.145714571457...
==> 9999z = 1457
==> z = 1457/9999
Then we end up with
x + y - z = 32345/369963
