Step 1: Understand the problem. The problem asks to find the square root (वर्गमूल) of the fraction $\frac{882}{1922}$. Step 2: Simplify the fraction $\frac{882}{1922}$ first. Find the prime factorization of the numerator, 882: $$ 882 = 2 \times 441 $$ $$ 441 = 21 \times 21 = 3^2 \times 7^2 $$ So, $882 = 2 \times 3^2 \times 7^2$. Find the prime factorization of the denominator, 1922: $$ 1922 = 2 \times 961 $$ $$ 961 = 31 \times 31 = 31^2 $$ So, $1922 = 2 \times 31^2$. Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2: $$ \frac{882}{1922} = \frac{882 \div 2}{1922 \div 2} = \frac{441}{961} $$ Step 3: Find the square root of the simplified fraction. $$ \sqrt{\frac{441}{961}} $$ This can be written as the square root of the numerator divided by the square root of the denominator: $$ \frac{\sqrt{441}}{\sqrt{961}} $$ Calculate the square root of the numerator: $$ \sqrt{441} = \sqrt{21^2} = 21 $$ Calculate the square root of the denominator: $$ \sqrt{961} = \sqrt{31^2} = 31 $$ Substitute these values back into the expression: $$ \frac{21}{31} $$ The square root of $\frac{882}{1922}$ is $\boxed{\frac{21}{31}}$.