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.
• n0 = 0.
• n1 = 1.
• nab = na ⋅ nb.
• na/b = na ÷ nb.
• 1a = a.
• nma = mna.
• nan = |a| if n is even; nan = a if n is odd.
• namn = |a|m if n is even; namn = am if n is odd.
• (mn)an = m|a| if n is even; (mn)an = ma if n is odd.
The overall theme of the last three rules is that roots and powers cancel, but the cancelled number leaves behind an absolute value if it was even. That said, you can ignore the absolute-value operation 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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