Use the square root key on a calculator and enter 3
Approximate it as 1.7320508076
Use the Babylonian method with a starting guess and repeat until accurate
Use Newton’s method to solve x² = 3
Use a table of squares and find the value between 1 and 2
Use the fact that 1.7² = 2.89 and 1.8² = 3.24 to narrow it down
Memorize the common approximation 1.732