1.

Comparison of surds.

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(a) If a and b are surds of the same order, say n, then \(\sqrt[n]{a}\) > \(\sqrt[n]{b}\) if a > b. 

 For example,  \(\sqrt[5]{27}\) > \(\sqrt[5]{20}\) as 27 > 20. 

(b) If the given surds are not of the same order, then first convert them to surds of the same order and then compare. For example, to compare \(\sqrt[3]{2}\) and \(\sqrt[4]{3}\), we take the LCM of the orders, i.e., 3 and 4, i.e., 12.

Now, \(\sqrt[3]{2}\)  = \(\sqrt[12]{2^4}\) = \(\sqrt[12]{16}\)             (∴ \(\sqrt[3]{2}\)  = \(2^\frac{1}{3}\) = \(2^\frac{4}{12}\))

  \(\sqrt[4]{3}\)  = \(\sqrt[12]{3^3}\) = \(\sqrt[12]{27}\)                   (∴  \(\sqrt[4]{3}\)  = \(3^\frac{1}{4}\) = \(3^\frac{3}{12}\))

Since, 27 > 16, therefore, \(\sqrt[12]{27}\) > \(\sqrt[12]{16}\), i.e, \(\sqrt[4]{3}\) > \(\sqrt[3]{2}\).


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