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Formulas and properties of nth root

The nth root of the number a is a number b, n-th power is equal to a
The following properties hold for nth root (a > 0, b > 0 and n, m, k is natural numbers):
  1. na b = na · nb

  2. $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

  3. $(\sqrt[n]{a})^k=\sqrt[n]{a^k}$

  4. $\sqrt[n]{\sqrt[m]{a}}=\sqrt[nm]{a}$

  5. $\sqrt[nk]{a^k}=\sqrt[n]{a}$

  6. $\sqrt[n·k]{a^{m·k}}=(\sqrt[n]{a^m})$

  7. $\sqrt[n]{a^n}=$ { |a|   if n – even
    a   if n – odd

  8. For any a and b, such that 0 ≤ ab truly following inequality:
    n a n b

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