Are you radical enough to solve this SURDOKU?

Question

All "surds" represent whole numbers smaller than 15, and the same number is never used twice with the same index. (It's a sudoku.)

Attribution: PUZZLEBOMB.co.uk. Puzzles by @stecks & @apaultaylor.

Answer 1

Surd-lution (sorry :P):

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To start:

Here is a table of the powers of 1-15 up to 5. We can grey out anything that isn't possible, whether that's due to too many digits, containing a 0 or duplicate digits:

Now the step by step:

1:

To start, the 5th root with 5 digits must be 32768, as that is the only 5th power that fits. This instantly means the 3rd root above must be 27, as that is the only 2 digit 3rd root ending in 7.

The 3 digit 5th root on the side must be 243, and the 4th root top left must be 16 as it can't be 81 due to the 8.

2:

Continuing on with deducing the roots, the 3 digit square root top right must be 196, and the square root top left can only contain, 4/5/9 so must be 49, completing the second row.

Further down, the 2 digit 5th root must be 32.

3:

In the top right box there is now only one place for a 3, and then only one place for the 4. This means the square root in the top middle box must just be simply 1, which then solves the top right box.

The remaining square root top left can now only be 36.

4:

Consider the overlapping 3rd roots in the middle. Running through the possibilities, the overlapping digit must be a 1 or 2, meaning the 'vertical' number is either 125 or 216. Either way, 1 and 2 are used, so the 'horizontal' number can't use them, but must have a 1 or 2 as the second digit. This only leaves 729 going across, and 216 going upwards.

This also means the other 3 digit 3-rd root further down can only be 512 or 729. But the last digit must be the first of a 2 digit square root, so it must be 512 and 25.

5:

The 3rd root on the left can now only be 64, meaning the square root below must be 9. The other single digit square root must be 4, and that leaves just two 2-digit square roots remaining.

The one middle left can only be 81, whereas the one bottom right can only be 16.

And just like that all the surds are completed:

6:

From here we can apply normal sudoku logic:

There is a naked single 5 in the left box, which in turn solves the 6th row. This leaves one place for a 7 in the right box, solving the 8th column. There is one place for a 9 in the top row, and only one place for a 2 and a 9 in the middle row. The right box can then be solved.

Finally everything now falls into place. First, the 7th column, then a 2 and a 1 can be placed in the first column, and a 9 in the 3rd. The 2s can then be completed, followed by the 4s, and then the 8s and 7s. Putting the final few numbers in we get the solution: