MATH or MAGIC!

Extrapolating on the principles used for cubing, to get the fourth power of a number (of two or more digits), simplify by splitting the number into two parts, a and b.

Thus (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

The solution comprises five parts, neatly fitting the five boxes shown above. Just adjust for excess carry over.

1. a⁴
2. a³ x b x 4
3. a² x b² x 6
4. b³ x a x 4
5. b⁴

A simple example to illustrate this technique.


23⁴ = 279841

The steps are:

1. 3⁴ = 81, put down the 1 in the rightmost box and carry over the 8
2. 3³ x 2 x 4 = 216, plus the 8 carried over is 224, put down the 4 and carry over the 22
3. 2² x 3² x 6 = 216, plus the 22 carried over is 238, put down the 8 and carry over the 23
4. 2³ x 3 x 4 = 96, plus the 23 carried over is 119, put down the 9 and carry over the 11
5. 2⁴ = 16, plus the 11 carried over, is 27 in the leftmost box

Another example.


108⁴ = 136048896

The steps are:

1. 8⁴ = 4096, put down the 6 in the rightmost box and carry over the 409
2. 8³ x 10 x 4 = 20480, plus the 409 carried over is 20889, put down the 9 and carry over the 2088
3. 10² x 8² x 6 = 38400, plus the 2088 carried over is 40488, put down the 8 and carry over the 4048
4. 10³ x 8 x 4 = 32000, plus the 4048 carried over is 36048, put down the 8 and carry over the 3604
5. 10⁴ = 10000, plus the 3604 carried over, is 13604 in the leftmost box

That's a hundred million-dollar figure worked out manually!