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09 Oct 2017, 9:28 pm
27 Oct 2017, 7:53 am
Do you mean without using a calculator? I think the best you can do is reduce the number of multiplications you have to do by factoring the exponent, remembering that multiplying two numbers with the same base is the same as just adding the exponents, e.g. (x ^ 2) * (x ^ 4) = x ^ 6. Obviously this is only easy to do by hand for integer exponents.
So if the problem is x ^ n, you work out x ^ 2. You can then get x ^ 4 by just squaring x ^ 2 which you've already worked out, etc. etc. For your example, you can reduce the number of multiplications needed for 9 ^ 27 from 26 down to 7 by doing:
Code:
9 ^ 2 = 9 * 9 = 81 (2 = 1 + 1)
9 ^ 4 = 81 ^ 2 = 6561 (4 = 2 + 2)
9 ^ 8 = 6561 ^ 2 = 43046721 (8 = 4 + 4)
9 ^ 16 = 43046721 ^ 2 = 1853020188851841 (16 = 8 + 8)
9 ^ 24 = 43046721 * 1853020188851841 = 79766443076872509863361 (24 = 16 + 8)
9 ^ 26 = 79766443076872509863361 * 81 = 6461081889226673298932241 (26 = 24 + 2)
9 ^ 27 = 6461081889226673298932241 * 9 = 58149737003040059690390169 (27 = 26 + 1)
9 ^ 4 = 81 ^ 2 = 6561 (4 = 2 + 2)
9 ^ 8 = 6561 ^ 2 = 43046721 (8 = 4 + 4)
9 ^ 16 = 43046721 ^ 2 = 1853020188851841 (16 = 8 + 8)
9 ^ 24 = 43046721 * 1853020188851841 = 79766443076872509863361 (24 = 16 + 8)
9 ^ 26 = 79766443076872509863361 * 81 = 6461081889226673298932241 (26 = 24 + 2)
9 ^ 27 = 6461081889226673298932241 * 9 = 58149737003040059690390169 (27 = 26 + 1)
Likewise for 256 ^ 5, you can get it down from 4 multiplications to 3 by realising 256 ^ 5 = ((256 ^ 2) ^ 2) * 256.
So it doesn't save much work for low exponents, but saves a lot for larger ones.
Still, I don't see why you'd want to do this by hand. If you meant something else, then just ignore this
.
28 Oct 2017, 11:09 am
03 Nov 2017, 11:36 am
