If the product of $n$ positive numbers is $1$, then their sum is

Since, $A.M. \geq G.M.$

let n positive numbers be $x_1, x_2,...,x_n$

$A.M. =\dfrac{x_1+x_2+...+x_n}{n}$

$G.M. = x_1*x_2*...*x_n$

Therefore,

$\dfrac{x_1+x_2+...+x_n}{n} \geq \sqrt[n]{x_1*x_2*...*x_n}$

$\implies x_1+x_2+...+x_n \geq n*\sqrt[n]{x_1*x_2*...*x_n}$

$\implies x_1+x_2+...+x_n \geq n$