Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be two natural numbers that have an equal number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
of digits in their decimal expansions. The first
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
digits (from left to right) of the numbers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
are equal. Prove that if
![m >\frac{n}{2},](/media/m/a/7/0/a70555eb27022b420e571fe8e7220d0f.png)
then
%V0
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$