The irrational numbers are the ones that can not be expressed as a fraction of two integers.
For example, pi or the square root of 2 are irrational numbers.
In set a, we have pi, that is irrational.
In set b, we have the square root of 80, that is a multiple of the square root of 5:
[tex]\sqrt[]{80}=\sqrt[]{16\cdot5}=\sqrt[]{16}\cdot\sqrt[]{5}=4\sqrt[]{5}[/tex]As the square root of 5 is irrational, then their multiples are also irrational.
In set c, we have the square root of 100, that have a rational solution.
[tex]\sqrt[]{100}=10[/tex]In set d, we have a periodic number. The periodic numbers, although having infinite decimals, can be expressed as fractions, so they are rational.
This set has all rational numbers.
Answers: set d and set c.
