Question:
Arrange in ascending order: \(\sqrt[3]{7}, \sqrt{4}, \sqrt[4]{10}\)
आरोही क्रम में व्यवस्थित कीजिए: \(\sqrt[3]{7}, \sqrt{4}, \sqrt[4]{10}\)
Options:
- \(\sqrt[4]{10}, \sqrt{4}, \sqrt[3]{7}\)
- \(\sqrt{4}, \sqrt[3]{7}, \sqrt[4]{10}\)
- \(\sqrt[3]{7}, \sqrt[4]{10}, \sqrt{4}\)
- \(\sqrt[4]{10}, \sqrt[3]{7}, \sqrt{4}\)
- \(\sqrt[4]{10}, \sqrt{4}, \sqrt[3]{7}\)
- \(\sqrt{4}, \sqrt[3]{7}, \sqrt[4]{10}\)
- \(\sqrt[3]{7}, \sqrt[4]{10}, \sqrt{4}\)
- \(\sqrt[4]{10}, \sqrt[3]{7}, \sqrt{4}\)
Correct Answer: d)
Explanation:
To compare roots with indices 3, 2, and 4, raise each to their LCM power (12):
\((\sqrt[4]{10})^{12} = 10^3 = 1000\)
\((\sqrt[3]{7})^{12} = 7^4 = 2401\)
\((\sqrt{4})^{12} = 4^6 = 4096\).
Since \(1000 < 2401 < 4096\), ascending order is \(\sqrt[4]{10}, \sqrt[3]{7}, \sqrt{4}\).
करणी की घात 3, 2, 4 का ल.स.प. 12 है। 12वीं घात लेने पर:
\((\sqrt[4]{10})^{12} = 1000\), \((\sqrt[3]{7})^{12} = 2401\), \((\sqrt{4})^{12} = 4096\)।
अतः आरोही क्रम \(\sqrt[4]{10}, \sqrt[3]{7}, \sqrt{4}\) है।