How to Evaluate √1061520150601 Without a Calculator?

[anemone]

Here is this week's POTW:

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Without the help of calculator, evaluate $\sqrt[6]{1061520150601}$.

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[anemone]

Congratulations to the following members for their correct solution:):

1. greg1313
2. Ackbach
3. lfdahl
4. RLBrown
5. kaliprasad

Solution from Ackbach:

Spoiler

The original number is $1061520150601\sim 10^{12}$. Hence, we expect the answer to be roughly $100$, and it must be greater than $100$. Now, we find that $110^2=12100$, and we can see immediately that the second digit in from the far left is only going to get greater and greater as we get up to the full sixth power. So the result must be less than $110$. The original number is odd, forcing the answer to be odd. The original number is not divisible by $5$, which rules out $105$. We still have $101, 103, 107,$ or $109$ as possible answers. The method of "casting out nines", or arithmetic modulo nine, does not rule out any of these possibilities, unfortunately. Note that $101^2=(100+1)(100+1)=10000+200+1=10201$. Then $101^4=10201^2=104060401$. From here it is not difficult to determine that $10201\cdot 104060401=1061520150601$, so the answer is $101$.

Alternate solution from kaliprasad:

Spoiler

$1061520150601$
= $1* 10^12 + 6 * 10^10 + 15 * 10^8 + 20 * 10^6 + 15 * 10^4 + 6 * 10^2 + 1$
= $(100 + 1)^6$ using binomial expansion
= $101^6$

so 6 th root of $1061520150601= 101$