Please make expressions equal to 6 using exactly four 4s

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Background: I am the source. I created this puzzle. This problem belongs in the Mathematical Puzzling section. If this is not the correct subforum, then I am asking it to be moved. People have already begun answering with solutions, and it would be nice for them to be able to try to complete the solutions.


You must use all four 4s.

You may use addition (+).

You may use subtraction (-).

You may use multiplication, such as with asterisks (*) and/or grouping symbols.

You may not use any division.

You may not use decimal points.

You may not use any concatenation.

You may use grouping symbols, such as parentheses and/or brackets.

You may not use any type of factorial signs.

You may use up to four square root symbols, including the use of nested square roots.

You may not use exponentiation.

You may not use logarithms.

You may not use trigonometric functions.

You may not use any other numbers, characters, or operations, to include no other roots.

This is in base 10.

Try to create and post 10 additional --> essentially <-- different solutions.

Note: Regarding "essentially different," $\sqrt4*\sqrt4$ is the same as $\sqrt{4*4 \ }$, for instance.

Also, $\sqrt{\sqrt{4*4 \ } \ }$ is the same as $\sqrt{\sqrt{4}}*\sqrt{\sqrt{4}}$.

Permutations are the same.


Here are examples:

$4 + 4 + \sqrt4 - 4 = 6$

$\sqrt4 + \sqrt4 - \sqrt4 + 4 = 6$

$4(4 - \sqrt{4}) - \sqrt{4} = 6$

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There are 2 best solutions below

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Some of the possible ways: $$4 + \sqrt{4} + 4 - 4 = 6$$ $$4 + \sqrt{4} + \sqrt{4} - \sqrt{4} = 6$$ $$4 + \sqrt{4} \cdot \sqrt{4} - \sqrt{4} = 6$$ $$4 + 4 - \sqrt{\sqrt{4}} \cdot \sqrt{\sqrt{4}} = 6$$

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\begin{align} &\sqrt{4\times 4\times 4}-\sqrt{4}\\ &\sqrt{4\times 4\times \sqrt{4}+4}\\ &\sqrt{4\times 4}+\sqrt{\sqrt{4\times 4}}\\ &\sqrt{\sqrt{4}}\times\sqrt{4\times 4+\sqrt{4}}\\ &\sqrt{4+\sqrt{4}}\times\sqrt{4+\sqrt{4}}\\ \end{align}