- #1
mahmoud2011
- 88
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Why the principal nth root is unique ? I mean in the definition of the princibal nth root it says:
let x be a real number and n is natural number [itex]n \geq 1[/itex].if n is odd , the principal nth root of x denoted [itex]\sqrt[n]{x}[/itex],is the unique real number satisfying [itex](\sqrt[n]{x})^{n} = x[/itex].if n is even , the principal nth root of x denoted [itex]\sqrt[n]{x}[/itex] is defined similarly provided [itex]x \geq 0[/itex] and [itex]\sqrt[n]{x} \geq 0[/itex].
so why it is unique is there a proof .
let x be a real number and n is natural number [itex]n \geq 1[/itex].if n is odd , the principal nth root of x denoted [itex]\sqrt[n]{x}[/itex],is the unique real number satisfying [itex](\sqrt[n]{x})^{n} = x[/itex].if n is even , the principal nth root of x denoted [itex]\sqrt[n]{x}[/itex] is defined similarly provided [itex]x \geq 0[/itex] and [itex]\sqrt[n]{x} \geq 0[/itex].
so why it is unique is there a proof .