Question - Compare Decimal and Fractional Numbers
Solution:
\[\sqrt{6.25} = \sqrt{25/4}\]
\[\sqrt{25/4} = 5 / 2\]
\[5/2 = 2.5\]
Таким образом, оба числа равны.
б) Сравним \(\sqrt{27}\) и \(\sqrt{28}\):\[\sqrt{27} \approx 5.196\]
\[\sqrt{28} \approx 5.292\]
\[\sqrt{27} < \sqrt{28}\]
в) Сравним 1.3 и 1.5:\[1.3 < 1.5\]
г) Сравним \(\frac{5}{\sqrt{5}}\) и \(\frac{6}{\sqrt{6}}\), преобразуя к десятичным дробям:\[\frac{5}{\sqrt{5}} = \frac{5}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{5 \sqrt{5}}{5} = \sqrt{5}\]
\[\frac{6}{\sqrt{6}} = \frac{6}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{6 \sqrt{6}}{6} = \sqrt{6}\]
\[\sqrt{5} \approx 2.236\]
\[\sqrt{6} \approx 2.449\]
\[\sqrt{5} < \sqrt{6}\]
д) Сравним 0.8 и 1:\[0.8 < 1\]
е) Сравним \(\sqrt{0.18}\) и 0.4:\[\sqrt{0.18} = \sqrt{\frac{18}{100}}\]
\[\sqrt{\frac{18}{100}} = \frac{\sqrt{18}}{10}\]
\[\sqrt{18} \approx 4.243\]
\[\frac{\sqrt{18}}{10} \approx 0.4243\]
\[0.4243 > 0.4\]
ж) Сравним \(\frac{4}{\sqrt{5}}\) и \(\frac{5}{\sqrt{6}}\), аналогично предыдущему пункту:\[\frac{4}{\sqrt{5}} = \frac{4}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{5}}{5}\]
\[\frac{5}{\sqrt{6}} = \frac{5}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{5 \sqrt{6}}{6}\]
\[4 \sqrt{5} \approx 8.944\]
\[5 \sqrt{6} \approx 12.247\]
\[\frac{8.944}{5} \approx 1.789\]
\[\frac{12.247}{6} \approx 2.041\]
\[1.789 < 2.041\]
з) Сравним \(\sqrt{3.5}\) и \(\frac{3.2}{\sqrt{3}}\):\[\sqrt{3.5} = \sqrt{\frac{7}{2}}\]
\[\sqrt{\frac{7}{2}} \approx 1.871\]
\[\frac{3.2}{\sqrt{3}} = \frac{3.2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3.2 \sqrt{3}}{3}\]
\[3.2 \sqrt{3} \approx 5.545\]
\[\frac{5.545}{3} \approx 1.848\]
\[\sqrt{3.5} > \frac{3.2}{\sqrt{3}}\]
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