Approximating the Geometric Mean

In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics. Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation \[ a \times 10^x \quad \text{and}\quad b \times 10^y, \] where the coefficients \(a\) and \(b\) are both between 1 and 9.