Find a rational number and an irrational number that are between 5.2 and 5.5. include the decimal approximation of the irrational number to the nearest hundredth.

To answer this question let us first define what a rational and irrational number is. A rational number is basically a number whose decimals are terminating (ie have an end) or repeating (continuously repeats the same pattern. 

An example of this from your question would be 

5.3 - terminating rational number
5.333333333333 - repeating rational number

Irrational numbers are basically numbers whose decimals are non terminating and non repeating. 

An example would be the square root of 29. When plugged in a calculator this would result in 5.3851648071 
This number goes on and on with no set pattern.