# 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.
• mnan = m|a| if n is even; mnan = ma if n is odd.
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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