Question

How to find whether x = 0.1010010001... + 0.272727... is an irrational number?

Answer 1

Notice that $0.272727\ldots=\dfrac{27}{99}=\dfrac{3}{11}$ is rational
and $0.1010010001\ldots$ is irrational as this is non-terminating and non-repeating.

Since sum of rational and irrational is always irrational, $x$ is irrational.