Suppose we want to use the Babylonian iterative method to find the first two estimates for the value of $\displaystyle \sqrt{{{86}}}.$ Because 86 is between $\displaystyle {9}^{{2}}$ and $\displaystyle {10}^{{2}}$, our first estimate will either be 9 or 10. However, since $\displaystyle {9}^{{2}}$ is closer to 86 than $\displaystyle {10}^{{2}}$, we take 9 to be our first estimate. Use this first estimate to find the second estimate for $\displaystyle \sqrt{{{86}}}.$

Second estimate = (This will normally be a fraction, not a whole number) Preview Question 6