√20=2√5 since √5 is irrational √20 is irrational...
√10=√(2*5) since both √2 and √5 are irrational, √10 is irrational...
√144=12 since 12 is equal to 12/1 this is a rational number...
.48 repeating...let x=.48 repeating and let 100x=48.48 repeating so:
100x-x=48.48-.48
99x=48
x=48/99
x=16/33 so 0.48 repeating (any repeating numerical pattern for that matter) is therefore a rational number.
Any number that can be expressed as a ratio of two integers is rational, if a number cannot be expressed as a ratio of two integers it is irrational.