The √ key on your calculator is broken. (So is the y^{½} 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 13^{2}=169 and 14^{2}=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 13.1^{2}=171.61 and 13.2^{2}=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 13.15^{2}=172.9225 and 13.16^{2}=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 ≈ 13.152…