Simplify the following: ((9-2)^2)^4/(4+3)^5 Multiply exponents. ((9-2)^2)^4 = (9-2)^(2×4): (9-2)^(2×4)/(4+3)^5 2×4 = 8: (9-2)^8/(4+3)^5 4+3 = 7: (9-2)^8/7^5 7^5 = 7×7^4 = 7 (7^2)^2: (9-2)^8/7 (7^2)^2 7^2 = 49: (9-2)^8/(7×49^2) | | 4 | 9 × | | 4 | 9 | 4 | 4 | 1 1 | 9 | 6 | 0 2 | 4 | 0 | 1: (9-2)^8/(7×2401) 7×2401 = 16807: (9-2)^8/16807 9-2 = 7: 7^8/16807 7^8 = (7^4)^2 = ((7^2)^2)^2: ((7^2)^2)^2/16807 7^2 = 49: (49^2)^2/16807 | | 4 | 9 × | | 4 | 9 | 4 | 4 | 1 1 | 9 | 6 | 0 2 | 4 | 0 | 1: 2401^2/16807 | | | 2 | 4 | 0 | 1 × | | | 2 | 4 | 0 | 1 | | | 2 | 4 | 0 | 1 | | 0 | 0 | 0 | 0 | 0 | 9 | 6 | 0 | 4 | 0 | 0 4 | 8 | 0 | 2 | 0 | 0 | 0 5 | 7 | 6 | 4 | 8 | 0 | 1: 5764801/16807 The gcd of 5764801 and 16807 is 16807, so 5764801/16807 = (16807×343)/(16807×1) = 16807/16807×343 = 343: Answer: | | 343
Simplify the following: ((9-2)^2)^4/(4+3)^5 Multiply exponents. ((9-2)^2)^4 = (9-2)^(2×4): (9-2)^(2×4)/(4+3)^5 2×4 = 8: (9-2)^8/(4+3)^5 4+3 = 7: (9-2)^8/7^5 7^5 = 7×7^4 = 7 (7^2)^2: (9-2)^8/7 (7^2)^2 7^2 = 49: (9-2)^8/(7×49^2) | | 4 | 9 × | | 4 | 9 | 4 | 4 | 1 1 | 9 | 6 | 0 2 | 4 | 0 | 1: (9-2)^8/(7×2401) 7×2401 = 16807: (9-2)^8/16807 9-2 = 7: 7^8/16807 7^8 = (7^4)^2 = ((7^2)^2)^2: ((7^2)^2)^2/16807 7^2 = 49: (49^2)^2/16807 | | 4 | 9 × | | 4 | 9 | 4 | 4 | 1 1 | 9 | 6 | 0 2 | 4 | 0 | 1: 2401^2/16807 | | | 2 | 4 | 0 | 1 × | | | 2 | 4 | 0 | 1 | | | 2 | 4 | 0 | 1 | | 0 | 0 | 0 | 0 | 0 | 9 | 6 | 0 | 4 | 0 | 0 4 | 8 | 0 | 2 | 0 | 0 | 0 5 | 7 | 6 | 4 | 8 | 0 | 1: 5764801/16807 The gcd of 5764801 and 16807 is 16807, so 5764801/16807 = (16807×343)/(16807×1) = 16807/16807×343 = 343: Answer: | | 343