Question

Consider a number a in the Reals with a>0 and n in the Naturals. Show that there exists a unique x in the Reals such that x^n=a

Answer 1

Conditions

Consider a number $a$ in the Reals with $a > 0$ and $n$ in the Naturals. Show that there exists a unique $x$ in the Reals such that $x^n = a$

Solution

$$x^{\infty} = \alpha$$

This is an arithmetical N-th root **by the definition**. A real number or complex number has $n$ roots of degree $n$.

$$x = \sqrt[n]{a}$$

This function has an unique value for all $a > 0$, that's why for each fixed $a$ exist one and only one $x$: $x = \sqrt[n]{a}$.