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3 to the power 3 to the power 3 translates into 3 to the power 27. Cyclicity is 3, 9, 7, 1 and to degree 4. So 3 to the power 27 goes for 3rd value in the cycle -7.

Step 4: Hence step 2 can be simplified as 27*27*...27 (9 27s)

Step 5: We know 27*27*27 ends in 3 (multiply it out. No need to do the entire multiplication. Only the last digits matter).

Step 6: Hence we end up abc3*abc3*abc3 = a number ending with 7 (again only the last digits matter)

Step 7: Voila!

This is way too complicated ! Every digit has a pattern that repeats. See the chart given below. Some repeat in multiples of 4, others repeat in multiples of 1 and 2. Key takeaway is that EVERY POWER REPEATS !

Powers of 2 - 2,4,8,6,,2,4,8,6 Powers of 3- 3,9,7,1,3,9,7,1 Powers of 4 - 4,6,4,6,4,6 Powers of 5 - 5,5,5,5,5,5,5, Powers of 6 - 6,6,6,6,6,6,6, Powers of 7 - 7,9,3,1,7,.. Powers of 8 - 8,4,2,6,8 Powers of 9 - 9,1,9,1

They key is to find the pattern for reptition. A units digit for a power of 3 repeats every 4 terms. The units digit for 3 ^ 27 ==> Hence the term that corresponds to Remainder of (27 divided by 4) = 3rd term = 7 is the answer

To stretch this a lil bit more 2 ^ 100 = Remainder of (100 divided by 4) = Remainder = 0 , ie 4th term

A word of caution - Focus on the word remainder. The division only tells us the POSITION of the term in the sequence and NOT the repeating term itself. 2^50 =-> REMAINDER OF (50 divided by 4) = 2ND TERM in the sequence, hence 4 -------------------------------- IT TAKES A LOT OF TIME AND EFFORT TO POST DETAILED RESPONSES YOUR KUDOS IS VERY MUCH APPRECIATED

Exponentiation is not associative. Addition and multiplication are. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2·3)·4 = 2·(3·4) = 24, but 23 to the 4 is 84 or 4,096, whereas 2 to the 34 is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up: