Simplifying roots

Here are the basic rules of algebra for simplifying roots. In these identities, a and b are arbitrary real numbers, while m and n are positive integers; however, the precise rule may depend on whether n is odd or even. In each rule, if the right-hand side is defined, then so is the left-hand side and the two sides are then equal. Of course, you can ignore the absolute values when a ≥ 0.

Here are some examples of taking roots:

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Here are some trickier examples that the book does using rational exponents, but I'll do them directly using rules for roots:

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