The greatest number among sqrt{5}, sqrt[3]{4} | vedclass

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Question

(C) To compare the numbers $\sqrt{5}, \sqrt[3]{4}, \sqrt[5]{2}, \sqrt[7]{3}$,we express them with a common index (root).The indices are $2, 3, 5, 7$.

Vedclass · Accepted Answer

$\sqrt{5}$

Vedclass · Answer

(C) To compare the numbers $\sqrt{5}, \sqrt[3]{4}, \sqrt[5]{2}, \sqrt[7]{3}$,we express them with a common index (root).The indices are $2, 3, 5, 7$. The Least Common Multiple $(LCM)$ of these indices is $2 	imes 3 	imes 5 	imes 7 = 210$.Now,convert each number to the $210^{th}$ root:$1$. $\sqrt{5} = 5^{1/2} = 5^{105/210} = \sqrt[210]{5^{105}}$$2$. $\sqrt[3]{4} = 4^{1/3} = 4^{70/210} = \sqrt[210]{4^{70}}$$3$. $\sqrt[5]{2} = 2^{1/5} = 2^{42/210} = \sqrt[210]{2^{42}}$$4$. $\sqrt[7]{3} = 3^{1/7} = 3^{30/210} = \sqrt[210]{3^{30}}$Comparing the radicands: $5^{105}$ is clearly the largest value among $5^{105}, 4^{70}, 2^{42}, 3^{30}$.Therefore,$\sqrt{5}$ is the greatest number.

The greatest number among $\sqrt{5}, \sqrt[3]{4}, \sqrt[5]{2}, \sqrt[7]{3}$ is

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