The greatest of $\sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}$ is

### Concept & Strategy Just like other root comparison problems, the core strategy here is to eliminate the radical signs. We achieve this by converting roots into fractional exponents and then multiplying all exponents by the Least Common Multiple (LCM) of their denominators. This scales the numbers proportionally, allowing for a straightforward integer comparison. $$\text{Compare } a^{\frac{1}{x}}, b^{\frac{1}{y}} \implies \text{Compare } a^{\frac{\text{LCM}}{x}}, b^{\frac{\text{LCM}}{y}}$$ ### Step-by-Step Solution * **Given:** The values to compare are $\sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}$. * **Calculation / Deduction:** 1. Express each root as a base with a fractional exponent: $$2^{\frac{1}{2}}, 3^{\frac{1}{6}}, 4^{\frac{1}{3}}, 5^{\frac{1}{4}}$$ 2. Extract the denominators from the exponents: 2, 6, 3, and 4. 3. Calculate the LCM of (2, 6, 3, 4): $$\text{LCM}(2, 6, 3, 4) = 12$$ 4. Raise every term to the power of 12 to transform them into whole numbers: * $(\sqrt{2})^{12} = (2^{\frac{1}{2}})^{12} = 2^6 = 64$ * $(\sqrt[6]{3})^{12} = (3^{\frac{1}{6}})^{12} = 3^2 = 9$ * $(\sqrt[3]{4})^{12} = (4^{\frac{1}{3}})^{12} = 4^4 = 256$ * $(\sqrt[4]{5})^{12} = (5^{\frac{1}{4}})^{12} = 5^3 = 125$ 5. Compare the simplified integer values: $256 > 125 > 64 > 9$. 6. The highest value is 256, which maps back to the original term $\sqrt[3]{4}$. ### Exam Strategy & Shortcut **Rapid Elimination:** You can compare them in pairs to save mental load. Compare $\sqrt{2}$ and $\sqrt[3]{4}$: Raise both to power 6. $2^3 = 8$ vs $4^2 = 16$. So $\sqrt[3]{4}$ is greater. Compare $\sqrt[6]{3}$ and $\sqrt[4]{5}$: Raise both to power 12. $3^2 = 9$ vs $5^3 = 125$. So $\sqrt[4]{5}$ is greater. Now just compare the winners: $\sqrt[3]{4}$ and $\sqrt[4]{5}$. Raise both to power 12. $4^4 = 256$ vs $5^3 = 125$. $\sqrt[3]{4}$ is the definitive winner. Breaking it down this way avoids juggling 4 large calculations at once. ### Common Pitfall A common trap is mixing up the order of the original roots when checking the options. Notice the question lists them in one order ($\sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}$) but the options shuffle them. Always explicitly map your final highest integer back to its exact initial radical form before selecting an option. ### Final Answer **Therefore, the correct answer is $\sqrt[3]{4}$.**