Respuesta:
To order the given radicals \(\sqrt{2}\), \(\sqrt[4]{3}\), and \(\sqrt[3]{4}\) from smallest to largest, the least common multiple (LCM) of their indices (2, 4, and 3) is calculated first, which is 12.
Using the LCM, each radical is rewritten as a twelfth root:
1. \(\sqrt{2} = \sqrt[2]{2} = \sqrt[12]{2^6} = \sqrt[12]{64}\)
2. \(\sqrt[4]{3} = \sqrt[12]{3^3} = \sqrt[12]{27}\)
3. \(\sqrt[3]{4} = \sqrt[12]{4^4} = \sqrt[12]{256}\)
Now, compare the twelfth roots:
- \(\sqrt[12]{27}\)
- \(\sqrt[12]{64}\)
- \(\sqrt[12]{256}\)
Ordering from smallest to largest:
1. \(\sqrt[4]{3} = \sqrt[12]{27}\)
2. \(\sqrt{2} = \sqrt[12]{64}\)
3. \(\sqrt[3]{4} = \sqrt[12]{256}\)
Thus, \(\sqrt[4]{3}\), \(\sqrt{2}\), and \(\sqrt[3]{4}\) are ordered from smallest to largest.
