This looks ugly, but don't panic.

For the first item, finding the 6th root of a square root is the same as finding the 12th root. We need to use this rule from before:

`rootmrootn(a) = root(mn)(a)`

So the first term is:

`root(6)(sqrt(2)) = root(6 times 2)(2) = root(12)(2)`

For the second term, we need to split up the `2^13` as follows:

`2^13= 2^12× 2`

We do this so that we end up with a "12th root of 2" term so that we can simplify the final answer. We also need the following identity for this part:

`root(n)(a^n) = ( rootn(a))^n = rootn((a^n)) = a`

We'll use this in the second line, to simplify the 12th root of `2^12`:


` = root(12)(2^12 times 2)`

`=root(12)(2^12) times root(12)(2)`

`=2 root(12)(2)`

Now let's put this all together and get the final answer:

`root(6)sqrt(2) - root(12)(2^13)`

` = root(12)(2) - 2 root(12)(2)`

`= - root(12)(2)`

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