Question

How can I find the exact distance between the points(√7,-√2)and (4√7,5√2)

Answer 1

$\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ \sqrt{7}}}\quad ,&{{ -\sqrt{2}}})\quad % (c,d) &({{4\sqrt{7}}}\quad ,&{{ 5\sqrt{2}}}) \end{array}~~~ % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}$

$\bf d=\sqrt{[4\sqrt{7}-\sqrt{7}]^2~+~[5\sqrt{2}-(-\sqrt{2})]^2} \\\\\\ d=\sqrt{(4\sqrt{7}-\sqrt{7})^2~+~(5\sqrt{2}+\sqrt{2})^2}\implies d=\sqrt{(3\sqrt{7})^2~+~(6\sqrt{2})^2} \\\\\\ d=\sqrt{3^2\cdot 7~~+~~6^2\cdot 2}\implies d=\sqrt{63+72}\implies d=\sqrt{135} \\\\\\ \begin{cases} 135=5\cdot 3\cdot 3\cdot 3\\ \qquad 5\cdot 3^2\cdot 3\\ \qquad 15\cdot 3^2 \end{cases}\implies d=\sqrt{15\cdot 3^2}\implies d=3\sqrt{15}$