Given x > 0 and a natural number n, it is true that there exists a unique positive real number r such that x^(1/n) = r. This is known as the nth root of x and is denoted as x^(1/n) = r.
The existence of this positive real number r is guaranteed by the fact that the nth power function, f(r) = r^n, is an increasing function over the positive real numbers, meaning that for any positive real numbers a and b, if a < b then f(a) < f(b). This implies that for any positive real number x, there exists a unique positive real number r such that f(r) = x.
To see this, consider the interval (0, x^(1/n)) for some x > 0 and natural number n. Since f(r) = r^n is increasing over the positive real numbers, it takes on all positive real values over this interval, including x. Therefore, there must exist a unique positive real number r in this interval such that f(r) = x, which means that x^(1/n) = r.
Therefore, for any positive real number x and natural number n, there exists a unique positive real number r such that x^(1/n) = r.