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When do I need absolute value when simplifying nth roots?

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In a radical, root expression with an exponent inside, on the radicand or a factor of the radicand, is equal to an even number index, then we need use absolute value in our answer answer. Possibly, this simplifies out, depending on other conditions. What are the rules and reasoning for this?

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Solution Summary

Explanation includes multiple examples where the absolute value persists and does not persist after simplification. Exaplanation includes a discussion of equivalent expressions, range, and domain.

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When do I need absolute values when simplifying nth roots?

This question arises typically when dealing with the principal square root which is the full name of the function involving the radical √x. Here the radicand, inside the radical, is x and the index is not stated and thus implied to be 2 for square root. An absolute value can arise from a simplification whenever the index is an even integer. Here are two cases, one when the absolute value is simplified out and one when it is required in the final answer.

Remember that the principal square root function can only access nonnegative values and produce nonnegative values, that is the function's domain ...

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