Yes, there is a method similar to one method for determining square roots.
You can also use logarithms to determine cube roots - more convenient than the algorithm, but still sort of longhand.
Cube root of 5 = 10^(log5/3)
Before calculators this was more of a longhand method since you had to get the logs from a table in your textbook and actually divide by 3 without a calculator.
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All numbers have cube roots (not necessarily integral cube roots) so every prime has cube roots.
Yes.
Many calculators will offer a way for you to do cube roots, or what number to the power of 3 will equal that number. There is no easy way to go about finding cube roots without a calculator, so I suggest not trying too hard.. but the cube root of 7 is 1.912931182772389 found from using a simple Cube Root calculator you can find on the internet.
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The answer depends on "different from WHAT?" Positive cube roots, or negative square roots?