以下は 9 * 9 * 9 * 9 の暗算の手順
9 * 9 = 81
81 * 9 = 720 + 9 = 729
729 * 9 = 700 * 9 + 29 * 9
= 6300 + 29 * (10 - 1)
= 6300 + 290 - 29
= 6300 + 290 - 30 + 1
= 6300 + 261
= 6561
または
729 * 9 = 729 * (10 - 1)
= 7290 - 729
= 7200 - 700 + 90 - 29
= 6500 + 90 - 30 + 1
= 6561
[追記]
または
81 * 81 = 81 * (80 + 1)
= 6480 + 81
= 6561
(80+1)^2 = 6400+160+1 かなあ