Is it possible to raise numbers to some natural power in columns? If not, why?

Question

I am wondering about something (sorry if the question might sound silly).

In order to perform multiplication in columns on the piece of paper, for example for 456 * 74, we don't have to perform addition in columns 74 times (74 times adding to the result the number 456) and instead we can use direct method of multiplication in columns by multiplying digits of the number by each others and adding the results at the end.

I wonder, if it is possible to do the same with powers? I know that to calculate 456 ^ 9, I can just 8 times multiply by 456 and get result. Or use shorter method and do ((456^2)^2)^2*456 to reduce the number of multiplications to 4 (however it is same with addition - for 456 * 9 we could do ((456*2)*2)*2+456 - but we don't do it - we multiply it directly instead).

I wonder if there is a workaround to do the powering directly on digits as it was possible with multiplication.

And if it isn't possible, has anybody proven why it is impossible?

Thank you in advance for answer! <3

Answer 1

In order to answer this question we need to understand real mathematics that is hidden under the hood. What you call "multiplication in columns" is based on the following mathematical facts:

1. Every natural number can be represented as $\sum a_i\cdot 10^i$, where $a_i\in\{0,1,2,3,4,5,6,7,8,9\}$. This is called base-$10$ representation and note that there is nothing special about $10$ here, everything will work the same if we switch to another number. As an example, when we write $234$, what we really mean is $4+3\cdot 10+2\cdot 100$.
2. Multiplication is distributive under addition, i.e. $a\cdot (b+c)=a\cdot b+a\cdot c$.

These two together also imply that given a base-$10$ representation, the multiplication by $10$ is the same as making shift and putting $0$ at the end, i.e.

$$1234\cdot 10=12340$$

The same in fact works for any base-$b$ representation. That's because

$$b\cdot(\sum a_i\cdot b^i)=\sum a_i\cdot b^{i+1}$$

This property is what is fundamental behind "column" representation.

Finally, the "multiplication in columns" follows from the observation that

$$x\cdot(\sum a_i\cdot 10^i)=\sum x\cdot a_i\cdot 10^i$$

Again, the choice of $10$ is not relevant.

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With all of that, we can see that the same won't work with taking power $x^y$. That's because this operation is not distributive over addition and also because taking (to the) power of $10$ (or any base) does not translate to shifting like with multiplication. The operation of taking power is just different.

The most similar formula is $a^{b+c}=a^b\cdot a^c$. There's also a formula for $(a+b)^c$, a.k.a. binomial theorem. However none of those are really similar to "multiplication in columns", those formulas don't translate easily to column representation.