9.565565556 = 9565565556/1000000000 = 2391391389/250000000

So it is rational.

However, if you mean 9.565565556... where it continues with one
more 5 than last time followed by a 6 and so on forever, then no,
it is not a rational.

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The decimal form of a rational number either terminates or
continues with the same one or more digits repeating, eg 1.5,
1.333..., 1.1818181..., 1.1666..., 1.1565656... are all rational
numbers.

If the decimal does not terminate nor continue forever with the
same repeating digit(s) then the number is irrational.

9.565565556 as written terminates and so it a rational
number.

9.565565556... does not terminate nor does it continue with the
same repeating digits (as an extra 5 is inserted before the next
6), so it it irrational.