## Using all digits from 1 to 9 once w/ (A/B)^{C} + (D/E)^{F} + (G/H)^{I}

$( \,A/B \,)^{C} \; + \; ( \,D/E \,)^{F} \; + \; ( \,G/H \,)^{I}$   ……….   (1)

The minimum value that can be obtained:

$(5/9)^{1} \; + \; (7/3)^{2} \; + \; (8/4)^{6} \; = \; 70$
$(5/9)^{1} \; + \; (8/4)^{6} \; + \; (7/3)^{2} \; = \; 70$
$(7/3)^{2} \; + \; (5/9)^{1} \; + \; (8/4)^{6} \; = \; 70$
$(7/3)^{2} \; + \; (8/4)^{6} \; + \; (5/9)^{1} \; = \; 70$
$(8/4)^{6} \; + \; (5/9)^{1} \; + \; (7/3)^{2} \; = \; 70$
$(8/4)^{6} \; + \; (7/3)^{2} \; + \; (5/9)^{1} \; = \; 70$

The maximum is :

$(4/2)^{5} \; + \; (6/3)^{7} \; + \; (8/1)^{9} \; = \; 134217888$
$(4/2)^{5} \; + \; (8/1)^{9} \; + \; (6/3)^{7} \; = \; 134217888$
$(4/2)^{7} \; + \; (6/3)^{5} \; + \; (8/1)^{9} \; = \; 134217888$
$(4/2)^{7} \; + \; (8/1)^{9} \; + \; (6/3)^{5} \; = \; 134217888$
$(6/3)^{5} \; + \; (4/2)^{7} \; + \; (8/1)^{9} \; = \; 134217888$
$(6/3)^{5} \; + \; (8/1)^{9} \; + \; (4/2)^{7} \; = \; 134217888$
$(6/3)^{7} \; + \; (4/2)^{5} \; + \; (8/1)^{9} \; = \; 134217888$
$(6/3)^{7} \; + \; (8/1)^{9} \; + \; (4/2)^{5} \; = \; 134217888$
$(8/1)^{9} \; + \; (4/2)^{5} \; + \; (6/3)^{7} \; = \; 134217888$
$(8/1)^{9} \; + \; (4/2)^{7} \; + \; (6/3)^{5} \; = \; 134217888$
$(8/1)^{9} \; + \; (6/3)^{5} \; + \; (4/2)^{7} \; = \; 134217888$
$(8/1)^{9} \; + \; (6/3)^{7} \; + \; (4/2)^{5} \; = \; 134217888$

135 distinct integer values can be formed from the expression (1) :

70, 132, 241, 356, 374, 407, 435, 593, 665, 672, 810, 882, 889,
1098, 1100, 2222, 2255, 2276, 2283, 2432,2824, 2942, 2948, 3937,
4154, 4507, 6526, 6740, 6908, 7307, 7713, 7913, 9205, 9373, 9772,
9967, 10219, 14465, 15762, 16880, 17323, 17346, 17521, 17540, 17575,
17738, 17792, 17816, 17920, 17923, 18068, 18137, 19936, 20054, 20058,
20723, 20984, 21902, 22314, 22977, 23188, 23432, 23449, 24241, 32090,
33408, 33625, 35019, 35036, 36498, 59181, 59204, 59433, 61244, 67966,
75449, 78198, 78641, 78664, 78858, 78893, 79110, 82302, 84750, 84767,
97816, 117690, 117896, 118148, 118754, 124242, 262515, 262897, 263296,
264363, 264577, 278560, 278967, 279977, 280183, 280435, 281041, 286529,
340285, 391265, 391482, 392876, 392893, 1679987, 1681835, 1953257, 1953280,
1953509, 1955320, 1969525, 2097459, 2097476, 2097696, 2097913, 4783005,
4783028, 4783220, 4783257, 4784009, 5765108, 5765125, 5765345, 5765562,
40353643, 40353666, 40353858, 40353895, 40354647, 43046881, and 134217888