Вычислите без округления значение арифметического корня с точностью до десятых, используя таблицу квадратов от \(\displaystyle 11\) до \(\displaystyle 19{\small :}\)
\(\displaystyle \sqrt{2}=\)\(\displaystyle ,\)\(\displaystyle \ldots\)
Представим число \(\displaystyle 2\) как дробь со знаменателем \(\displaystyle 100{\small :}\)
\(\displaystyle 2=\frac{200}{100}{\small.}\)
Тогда
\(\displaystyle \sqrt{2}=\sqrt{\frac{200}{100}}=\frac{\sqrt{200}}{10}{\small.}\)
Найдем в нижней строке таблицы квадратов меньшее и большее числа, ближайшие к \(\displaystyle 200 {\small ,}\) – это \(\displaystyle \bf 196 \) и \(\displaystyle \bf 225{\small :}\)
| \(\displaystyle \color{blue}{11^2}\) | \(\displaystyle \color{blue}{12^2}\) | \(\displaystyle \color{blue}{13^2}\) | \(\displaystyle \color{red}{14^2}\) | \(\displaystyle \color{red}{15^2}\) | \(\displaystyle \color{blue}{16^2}\) | \(\displaystyle \color{blue}{17^2}\) | \(\displaystyle \color{blue}{18^2}\) | \(\displaystyle \color{blue}{19^2}\) |
| \(\displaystyle \bf \color{blue}{121}\) | \(\displaystyle \bf \color{blue}{144}\) | \(\displaystyle \bf \color{blue}{169}\) | \(\displaystyle \bf \color{red}{196}\) | \(\displaystyle \bf \color{red}{225}\) | \(\displaystyle \bf \color{blue}{256}\) | \(\displaystyle \bf \color{blue}{289}\) | \(\displaystyle \bf \color{blue}{324}\) | \(\displaystyle \bf \color{blue}{361}\) |
Тогда можно записать неравенство:
\(\displaystyle 196 < 200 < 225{\small ,}\)
\(\displaystyle 14^2 < 200 < 15^2{\small ,}\)
\(\displaystyle \sqrt{14^2}<\sqrt{200}<\sqrt{15^2}{\small ,}\)
\(\displaystyle 14<\sqrt{200}<15{\small .}\)
Разделим неравенство на \(\displaystyle 10{\small :}\)
\(\displaystyle \frac{14}{10}<\frac{\sqrt{200}}{10}<\frac{15}{10}{\small,}\)
\(\displaystyle 1{,}4<\frac{\sqrt{200}}{10}<1{,}5{\small,}\)
\(\displaystyle 1{,}4<\sqrt{2}<1{,}5{\small.}\)
Таким образом,
\(\displaystyle \sqrt{2}=1{,}4\ldots\)
Ответ: \(\displaystyle \sqrt{2}=1{,}4\ldots\)