What is taking the square root of a number actually do?

Now I know that taking the square root of a number is just finding another number that when squared will give the initial number, like how the square root of 9 is 3 because 3^2 is 9.

BUT we can verify that 3^2 is 9 because we can multiply 3 by 3 and get 9, so my question is; Is there like a similar method for finding the square root of a number?

If a^2 = a x a then is there a similar formula for √a?