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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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Q: Is there a longhand method for finding cube roots?
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