Question 319471
<pre><b>
Does "3^3^3" or {{{drawing(50,50,-1,1,-1,.5, locate(-1,0, 3^3^3) )}}} mean {{{(3^3)^3=19683}}} or {{{drawing(50,50,-1,1,-1,.5, locate(-1,0, 3^((3^3))) )}}}?


I will assume it's {{{drawing(50,50,-1,1,-1,.5, locate(-1,0, 3^((3^3))) )}}}; otherwise we'd just do it with the calculator
{{{(3^3)^3=19683}}} and the answer would obviously be 3.

So I assume you mean:

{{{drawing(50,50,-1,1,-1,.5, locate(-1,0, 3^((3^3))) )}}}

3<sup>0</sup> = 1
3<sup>1</sup> = 3
3<sup>2</sup> = 9
3<sup>3</sup> = 27

3<sup>4</sup> = 81
3<sup>5</sup> = 243
3<sup>6</sup> = 729
3<sup>7</sup> = 2187

We can see that the last digits repeat in cycles of 4:  1,3,5,7,1,3,5,7,...
So the last digit of {{{3^n}}} is the same as the last digit of n mod 4.

Therefore since {{{drawing(50,50,-1,1,-1,.5, locate(-1,0, 3^((3^3))) )}}} is the same as {{{3^27}}}, and since 27 mod 4 = 3, 
the last digit of it is the same as the last digit of {{{3^3=27}}}, so the answer is 7. 

Edwin</pre>