ANSWERS: 3
  • Well, you can do it in your head just by multiplying 3 by 3, eight times, and each time you do the multiplication, you drop the upper digits since they aren't important in determining the units digit. For example: units(3^0) = 1 units(3^1) = 3 units(3^2) = units(3 * 3) = 9 units(3^3) = units(3 * 9) = units(27) = 7 units(3^4) = units(3 * 7) = units(21) = 1 units(3^5) = units(3 * 1) = 3 units(3^6) = units(3 * 3) = 9 units(3^7) = units(3 * 9) = units(27) = 7 units(3^8) = units(3 * 7) = units(21) = 1 See how it starts to repeat? You could generalize the rule for units(3^x) using this pattern, but that's more than you asked for, so I will stop here.
  • To get the answer to this, first multiply 3 by itself 8 times (e.g. 3x3x3x3x3x3x3x3=?). What is the last digit in this number? This is the digit that gives the value of the ones. The ones digit is always the digit immediately to the left of the decimal point.
  • 3^8 is (3^2)^4 which is 9^4 9^4 is (9^2)^2 which is 81^2 81^2 is 80*80 + 80*1 + 1*80 + 1*1 = 6400 + 80 + 80 + 1 = 6561 OR you could note that the rest of the digits except the ones digit don't make a difference: so 3^1 = 1*3 3^2 = 3*3 = 9, 3^3 = 9*3 = 27 = ???7 3^4 = ???7*3 = ???1 so multiply anything by four lots of 3 and the last digit is multiplied by 1 so 3^8 = 1*3^4*3^4 ends with 1.

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