What is the largest $n$-digit number which is also an exact $n$th power?

For example,the largest $2$-digit number which is an exact $2$nd power is $81$.


You want the largest $a^n$ such that $10^{n-1}\le a^n\lt 10^n$ for a given $n$.

From the right inequality, we have $a=9$.

Now $10^{n-1}\le 9^n$ is true only when $n\le 21$.

So, the answer is $9^n$ for $n=1,2,\cdots, 21$. For $n\ge 22$, there is no such number.

  • 2
    $\begingroup$ FWIW, $9^{22}$ is 984770902183611232881, which has 21 digits. $\endgroup$ – PM 2Ring Oct 11 '15 at 11:30

The trick is to rearrange the problem: rather than ask for the largest $n$-digit number that is an $n$-th power, instead ask for the largest number whose $n$-th power is an $n$-digit number.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.