Mentally raising small numbers to small powers

First, students will mentally evaluate small whole numbers to whole number powers \(\le 3.\) For example, evaluate \(4^3.\) If students have memorized \(4^2,\) then \(4^3\) is \(4^2 \cdot 4 = 16 \cdot 4.\) Because I don't know \(16 \cdot 4\) off the top of my head, I break it down into \((10 + 6) \cdot 4,\) which I mentally distribute to obtain \(40 + 24,\) which is easy to add mentally to get \(64.\) So \(4^3 = 64.\) Next, challenge your students to mentally evaluate \(0^4, 1^4, 2^4, 3^4, 4^4, 5^4,\) one at a time, in that order. To find \(5^4\) I think \(25 \cdot 25.\) Since I can't do \(25 \cdot 25\) in my head, I break it up into \(25(20 + 5).\) Since I can't easily do \(25 \cdot 20,\) I think \(25 \cdot 10 \cdot 2,\) so doubling \(250\) gives me \(500.\) I remember I must still find \(25 \cdot 5.\) If you think about quarters, five quarters is \($1.25,\) so \(25 \cdot 5\) must be \(125.\) Adding this to \(500\) gives me \(625.\) So \(5^4 = 625.\) There's lots of different ways you could find \(5^4.\) Let students figure out their own strategies. Then have students explain their strategies orally. They probably won't recall exactly what they did, so you'll have to ask probing questions as they attempt to communicate their strategy.