the pattern in math powers

perfect squares, cubes and numbers raised to other powers get big fast and confuse tons of people. and there are so many of them — an infinite number.

a pattern makes them a little easier. here’s what’s up:

1. perfect squares end in only five digits: 1, 4, 5, 6 and 9. if it doesn’t end in one of those, it’s not a perfect square. for example, 1²=1, 9²=81, 2²=4, 8²=64.
2. within that 1–4–5–6–9 pattern is another: the numbers whose perfect squares end in those numbers add up to 10. here’s a chart, with ending digits bolded:

1 (1²=1) + 9 (9²=81)=10

2 (4) + 8 (64) = 10

3 (9) + 7 (49) = 10

4 (16) + 6 (36) = 10

5 and 10 don’t quite fit this, but 5 x 2 = 10, and 10 is already 10.

3. perfect cubes have a pattern too. here’s another chart, with ending digits again bolded:

1 (1³=1) matches itself.

2 (2³=8) ends in 8. 8 (8³=512) ends in 2. again, 2+8=10.

3 (3³=27) ends in 7. 7 (7³=343) ends in 3. again, 3+7=10.

4 (4³=64) matches itself.

5 (5³=125) matches itself.

6 (6³=236) matches itself.

7 and 8 are done above.

9 (9³=729) matches itself.

and again, 10 (10³=1,000) matches itself. so the pattern here is match, pair pair, match, match, match, pair, pair, match. 1, 2, 3, 2, 1.

4. perfect fourth power numbers with a nonzero final integer end in either 1 or 6 (except 5, which always ends in 5):

1 (1⁴=1) is 1.

2 (2⁴=16) is 6.

3 (3⁴=81) is 1.

4 (4⁴=256) is 6.

5 (5⁴=625) is 5.

6 (6⁴=1296) is 6.

7 (7⁴=2401) is 1.

8 (8⁴=4096) is 6.

9 (9⁴=6561) is 1.

5: everything repeats:

a) the identity property of multiplication tells us that any number times 1 is itself, so those odd numbers up there get multiplied by 1, giving themselves. but where this gets fun is that any nonzero even integer times 6 equals a number that ends in that integer, which gives us …

1 (1⁵=1) is 1

2 (2⁵=32) is 2.

3 (3⁵=243) is 3.

4 (4⁵=1024) is 4.

5 (5⁵=3125) is 5.

6 (6⁵=7776) is 6.

7 (7⁵=16807) is 7.

8 (8⁵=32768) is 8.

9 (9⁵=59049) is 9.

and since we’re now back to all the numbers’ original final integers after adding four powers (going from a power of 1 to a power of 5), we can conclude that this pattern repeats every four powers. so 2¹ ends in 2, as does 2⁵ (32), 2⁹ (512), 2¹³ (8192), etc.

some of these numbers do repeat faster, mind: 0, 1, 5 and 6 always end in 0, 1, 5 and 6, respectively, and 9 and 4 alternate between 9 and 1 and 4 and 6, respectively. but overall, this pattern can help you memorize and understand powers, plus tell at a glance if a number can be a perfect square, cube, etc., plus what integer its base might end in.

good luck <3