Answer:
Considering first question
     Generally the coefficient of performance of the air condition  is mathematically represented as
    
Here  is the inside temperature
 is the inside temperature 
 while   is the outside temperature
 is the outside temperature 
What this coefficient of performance represent is the amount of heat the air condition can remove with 1 unit of electricity 
So it implies that the air condition removes    heat with 1 unit of electricity
 heat with 1 unit of electricity 
 Now from the question we are told that the rate at which heat enters an air conditioned building is often roughly proportional to the difference in temperature between inside and outside. This can be mathematically represented as 
          
=>        
Here k is the constant of proportionality 
So  
     since  1 unit of electricity  removes    amount of heat
  amount of heat 
    E  unit of electricity will remove  
So 
        
 
=>   
given that   is constant
 is constant
     =>  
From this above equation we see that the  electricity required(cost of powering and operating the air conditioner) is approximately proportional to the square of the temperature difference.
  Considering the  second question 
Assuming that  
  and      
Hence  
      
Here K stand for a constant 
So  
         
=>      
Now if  the  
Then 
        
=>      
So  from this see that the electricity require (cost of powering and operating the air conditioner)when the inside temperature is low  is  much higher than the electricity required when the inside temperature is higher
 Considering the  third question
Now in the case where the  heat that enters the building is at a rate proportional to the square-root of the temperature difference between inside and outside 
We have that 
        
So 
        
=>   
Assuming  is a constant
 is a constant 
Then  
      
From this above equation we see that the  electricity required(cost of powering and operating the air conditioner) is approximately proportional to the square root  of the cube of the  temperature difference.