Simple representation of radicals of a number

Question

I'm trying to represent radicals of a number in an easily understood, programmable, and calculate-able way. For instance: multiplication is a string of additions, division is a count of subtractions, exponentiation is a string of multiplications, and logarithms are a count of divisions. Following these ideas, is there any way to think of calculating radicals this way. For simplicity, let's assume that we're only working with integers for all of these operations.

Answer 1

Exponentiation

Informally, we can say that exponentiation has two inverses. Let $n,b,p\in\mathbb{N}$.

$$\underbrace{p=\log_{b} n}_{\text{logarithm}}\quad\Longleftrightarrow\quad n=b^{\,p}\quad\Longleftrightarrow\quad\underbrace{b=\sqrt[p]{n}}_{\text{root}}$$

We see from here that a logarithm essentially is a power and likewise a root is a base, with respect to the original exponential expression $b^{\,p}$.

Having learned this, let's consider what $n=b^{\,p}$ means in the iterative sense for whole numbers.

$$n=b\underbrace{\times \cdots\times b}_{p-1\;\text{times}}$$

Notice that the exponential expression can be written as a product of $p$-many copies of $b$, or alternatively $b$ multiplied by itself $p-1$ times.

Logs and Radicals

It turns out that both inverse operations involve counted division. Notice from the above definition that we can derive

$$b=n\underbrace{\div b\div\cdots\div b}_{p-1\;\text{times}}$$

which makes sense, since exponentiation is iterated multiplication.

The distinction between logarithms and radicals lies in what sort of numbers we know and can work with: the logarithm tells how many times to divide through while the root indicates what number to divide through by.

In other words, think of $p=\log_b n$ as the count of the division and $b=\sqrt[p]{n}$ as the divisor itself.

It should be emphasized that the counted division approach cannot directly compute either the root or logarithm (unsurprisingly, as neither does counted subtraction for division). Regardless, it gives us a way of understanding these inverse operations iteratively.

Conclusion

For sake of example, observe that $125=5^3$.

How many times do we have to divide $125$ by $5$ to end up with $5$? Twice. This plus one is the logarithm, i.e. $\log_5 125=3$.

What number do we have to divide $125$ by $3-1$ times (twice) to get that same number? This is a bit trickier, but trial-by-guessing gives us the root, i.e. $\sqrt[3]{125}=5$.

The above ideas comprise a neat conceptual approach, but when actually trying to evaluate radicals and logs, I defer to the original exponential relationship. If I want to find $\log_5 125$, I ask "5 to what power is 125?". Likewise, for $\sqrt[3]{125}$ I would inquire "What base cubed equals 125?"