Answer
$$7.190$$
Work Step by Step
$$\eqalign{
& {\text{We can note that }}\sqrt {51.7} {\text{ is a solution of the equation }}{x^2} - 51.7 = 0.{\text{ }} \cr
& {\text{Let }}f\left( x \right) = {x^2} - 37.6,\,\,\,\,\,\,f'\left( x \right) = 2x \cr
& \cr
& {\text{Since 7 < }}\sqrt {51.7} < 8,{\text{ we can use }}{c_1} = 7{\text{ as the first approximation}}{\text{. }} \cr
& {\text{Then using Newton's Method we have}}: \cr
& {c_2} = {c_1} - \frac{{f\left( {{c_1}} \right)}}{{f'\left( {{c_1}} \right)}} = 7 - \frac{{{{\left( 7 \right)}^2} - 51.7}}{{2\left( 7 \right)}} = 7.1928 \cr
& {c_3} = {c_2} - \frac{{f\left( {{c_2}} \right)}}{{f'\left( {{c_2}} \right)}} = 7.1928 - \frac{{{{\left( {7.1928} \right)}^2} - 51.7}}{{2\left( {7.1928} \right)}} = 7.1902 \cr
& {\text{In the same way}} \cr
& {c_4} = 7.1902 - \frac{{{{\left( {7.1902} \right)}^2} - 51.7}}{{2\left( {7.1902} \right)}} = 7.1902 \cr
& \cr
& {\text{Since }}{c_3} = {c_4} = 7.1902,{\text{ to the nearest thousand}}{\text{, }}\sqrt {51.7} = 7.190 \cr} $$
