Is this a new theorem
If $(n, m, k,)$ are three positive integers, where, ($k > 1$), $p$ is prime number satisfying the following inequality identity, $$m^k < p*(10)^{kn} < (m + 1)^k$$
Then, $m$ is an integer that represents the sequence digits of $k’th$ real arithmetical root of $p$ with $n$ number of accurate digits after the decimal notation
Example
What is the cubic real arithmetical root of (7), $\sqrt[3]{7}$ to five digits of accuracy after the decimal notation?
Here, we have (k = 3), (p = 7), (n = 5), then (m) can be obtained from the following inequality identity:
$$m^3 < 7*(10)^{15} < (m+1)^3$$, where $ (m = 191293)$, just by too little inspection
And the real arithmetical cubic root of (7), $\sqrt[3]{7}$ is approximately (1.91293) to five digits of accuracy after the decimal notation as required
Any constructive comments are welcomed
It is not necesary that $p$ is prime here. I fact, for arbitrary positive $p$, $$m^n<p\cdot10^{kn}<(m+1)^n $$ is equivalent to (just take $n$th roots) $$ m<\sqrt[n]p\cdot 10^k<m+1$$ and after division by $10^k$ to $$\frac m{10^k}< \sqrt[n]p<\frac{m+1}{10^k}$$