section_path: Identities and properties
section_name: Identities and properties
paragraph_text: Identities and properties
  
  Expressing the degree of an n th root in its exponent form, as in x 1 / n {\displaystyle x^{1/n}}, makes it easier to manipulate powers and roots. If a {\displaystyle a} is a non-negative real number,
  
  a m n = (a m) 1 / n = a m / n = (a 1 / n) m = (a n) m. {\displaystyle {\sqrt{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt{a}})^{m}.}
  
  Every non-negative number has exactly one non-negative real n th root, and so the rules for operations with surds involving non-negative radicands a {\displaystyle a} and b {\displaystyle b} are straightforward within the real numbers:
  
  a b n = a n b n a b n = a n b n {\displaystyle {\begin{aligned}{\sqrt{ab}}&={\sqrt{a}}{\sqrt{b}}\\{\sqrt{\frac {a}{b}}}&={\frac {\sqrt{a}}{\sqrt{b}}}\end{aligned}}}
  
  Subtleties can occur when taking the n th roots of negative or complex numbers. For instance:
  
  − 1 × − 1 ≠ − 1 × − 1 = 1, {\displaystyle {\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad }
  
  but, rather,
  
  − 1 × − 1 = i × i = i 2 = − 1. {\displaystyle \quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.}
  
  Since the rule a n × b n = a b n {\displaystyle {\sqrt{a}}\times {\sqrt{b}}={\sqrt{ab}}} strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.