Comparison of Surds

In comparison of surds we will discuss about the comparison of equiradical surds and comparison of non-equiradical surds.

I. Comparison of equiradical surds:

In case of equiradical surds (i.e., surds of the same order) \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\), we have \(\sqrt[n]{a}\) > \(\sqrt[n]{b}\) when x > y.

For example,

(i) √5 > √3, since 5 > 3

(ii) ∛21 < ∛28, since 21 < 28.

(iii) ∜10 > ∜6, since 10 > 6.

 

II. Comparison of non-equiradical surds:

In case of comparison between two or more non-equiradical surds (i.e., surds of different orders) we express them to surds of the same order (i.e., equiradical surds). Thus, to compare between ∛7 and ∜5 we express them to surds of the same order as follows:

Clearly, the orders of the given surds are 3 and 4 respectively and LCM Of 3 and 4 is 12.

Therefore, ∛7 = 7\(^{1/3}\) = 7\(^{4/12/}\) = \(\sqrt[12]{7^{4}}\) = \(\sqrt[12]{2401}\) and

∜5 = 5\(^{1/4}\) = 5\(^{3/12}\) = \(\sqrt[12]{5^{3}}\) = \(\sqrt[12]{125}\)

Clearly, we see that 2401 > 125

Therefore, ∛7 > ∜5.


Example of comparison of surds:

Convert each of the following surds into equiradical surds of the lowest order and then arrange them in ascending order.

                                   ∛2, ∜3 and \(\sqrt[12]{4}\)

Solution:

∛2, ∜3 and \(\sqrt[12]{4}\)

We see that the orders of the given surds are 3, 4 and 12 respectively.

Now we need to find the lowest common multiple of 3, 4 and 12.

The lowest common multiple of 3, 4 and 12 = 12

Therefore, the given surds are expressed as equiradical surds of the lowest order (i.e. 12th order) as follows:

∛2 = 2\(^{1/3}\) = 2\(^{4/12}\) = \(\sqrt[12]{2^{4}}\) = \(\sqrt[12]{16}\)

∜3 = 3\(^{1/4}\) = 3\(^{3/12}\) = \(\sqrt[12]{3^{3}}\) = \(\sqrt[12]{27}\)

\(\sqrt[12]{4}\) = 4\(^{1/12}\) = \(\sqrt[12]{4^{1}}\) = \(\sqrt[12]{4}\)

Therefore, equiradical surds of the lowest order ∛2, ∜3 and \(\sqrt[12]{4}\) are \(\sqrt[12]{16}\), \(\sqrt[12]{27}\) and \(\sqrt[12]{4}\) respectively.

Clearly, 4 < 16 < 27; hence the required ascending order of the given surds is:

\(\sqrt[12]{4}\), ∛2, ∜3





11 and 12 Grade Math

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