The options are analyzed as follows:
1: The overline 0.729 means:
[tex]0.729729729\ldots[/tex]
This is a non- terminating recurring decimal hence it is rational.
2: The nuumber 0.0195347... is a non terminating and non recurring decimal hence it is irrational.
3. The number -12 is rational since it is an integer.
4.The number 4.565656... is a non terminating recurring decimal hence it is rational.
5. The number given by:
[tex]\sqrt[]{18}=3\sqrt[]{2}=4.242640687\ldots[/tex]
Since square root of 2 is irrational and 3 is rational the product is irrational.
Also the decimal is non terminating non recurring hence it is irratonal.
So the options 2 and 5 are irrational.
