Official Solution:What is the last digit of \(3^{3^3}\)?

A. 1

B. 3

C. 6

D. 7

E. 9

THEORY If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:

\(a^{m^n}=a^{(m^n)}\) and not \((a^m)^n\), which on the other hand equals to \(a^{mn}\).

So:

\((a^m)^n=a^{mn}\);

\(a^{m^n}=a^{(m^n)}\) and not \((a^m)^n\).

BACK TO THE QUESTION According to the above: \(3^{3^3}=3^{(3^3)}=3^{27}\)

Next, the units digit of 3 in positive integer power repeats in pattern of 4: {3, 9, 7, 1}. So, the units digit of \(3^{27}\) (27=

4*6+

3) is the same as the units digit of \(3^3\), which is 7.

Answer: D

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