(10 points) The radix economy of a positive integer n and a base b is equal to b times the number of numerals it would take in base b to represent n. For example, if the number n=35 and b=10, then we need 2 numerals to represent n then we multiply by b=10 since there are 10 possible digits. So in this example, the radix economy of 35 in base 10 is 210=20. Let's look at this same number n=35 with base b=4. Then 35=(203)4 requires 3 numerals. So the radix economy in this case is 34=12 which is lower than 20 (the radix economy in base 10). The idea is that for higher bases, the number of numerals needed is fewer than for lower bases but higher bases require more numerals to remember. (One application of this concept is designing a telephone menu system for many options. Each layer of the menu has a particular number of choices that the customer must listen to before making a selection. If there are more choices per layer, then each layer takes a long time but if there are fewer choices per menu then there are more layers to listen to. Striking a balance to minimize the average time for a person to spend on the phone is an application of radix economy.) In general, for some n and some base b2, the function f(n,b) is the radix economy of n with base b defined as: f:Z+Z2N f(n,b)=blogb(n)+1 Exercise 1: (1 point, for fair effort completeness) Explain why f computes the radix economy of n with base b Exercise 2: (4 points) For each base in the set {2,3,4,5,6,7,8,9}, compute the radix economy of 800 . (show your work) Which base gives the lowest radix economy? Exercise 3: (4 points) For each base in the set {2,3,4,5,6,7,8,9}, compute the radix economy of 700 . (show your work) Which base gives the lowest radix economy? Exercise 4: ( 1 point for fair effort completeness:) Is there a particular base that will always give the optimal radix economy? How does this relate to computer design? Is base 2 the "best" base to be using? why or why not