Digits and Powers

Converting all the roots to powers yields:

We can multiply these powers together to obtain a single power:

$10\times \frac{1}{100}\times 1,000\times \frac{1}{10,000}\times 100,000\times \frac{1}{1,000,000}\times 10,000,000$

= $(10\times \frac{1}{100})\times (1,000\times \frac{1}{10,000})\times (100,000\times \frac{1}{1,000,000})\times 10,000,000$

= $\frac{1}{10}\times \frac{1}{10}\times \frac{1}{10}\times 10,000,000$

= $\frac{1}{1,000}\times 10,000,000$

=10,000

Thus, the expression simplifies to $10^{10,000}$ . This is equivalent to $1$ followed by $10,000$ $0$ s: that is, a $10,001$ digit number. So the solution is $\boxed{10,001}$ .

QED.