The √  key on your calculator is broken. (So is the y½ subroutine.) How can you find √173 to 3 decimal places using your calculator and nobody else’s?

First we need to identify the closest integer values to √173. Since we know 132=169 and 142=196, we can state:

169 < √173 < √196 or
13 < √173 < 14 Knowing that 169 is closer to 173 than 196, we can deduce that: 13.0 < √173 < 13.5
Now examine the values between 13 and 13.5.

We see that 13.12=171.61 and 13.22=174.24 suggesting

171.61 < √173 < √174.24
That is, 13.1 < √173 < 13.2 Notice that 174.24 is closer to 173 than 171.61. This suggests that we should examine values closer to 13.2 than 13.1.
We see that 13.152=172.9225 and 13.162=173.1856, suggesting

172.9225 < √173 < √173.1856
That is, 13.15 < √173 < 13.16 If we continue to carry out this process we see that
13.152 < √173 < 13.153

Thus, √173 &#8776 13.152…
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