**Definitions of surds:** A root of a positive real quantity is called a surd if its value cannot be exactly determined. It is a number that can’t be simplified to remove a square root (or cube root etc). For example, each of the quantities √3, ∛7, ∜19, (16)^2/5 etc. is a surd.

More Examples:

- √2 (square root of 2) can’t be simplified further so it is a surd
- √4 (square root of 4) CAN be simplified to 2, so it is NOT a surd

**Comparison of Surds**

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

**Comparison of equiradical surds:**

In case of equiradical surds (i.e., surds of the same order) ^{n}√a and ^{n}√b, we have ^{n}√a > ^{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.

**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} = ^{12}√7^{4} = ^{12}√2401 and

∜5 = 5^{1/4} = 5^{3/12} = ^{12}√5^{3} = ^{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 ^{12}√4

Solution:

∛2, ∜3 and ^{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} = ^{12}√2^{4} = ^{12}√16

∜3 = 3^{1/4} = 3^{3/12} = ^{12}√3^{3} = ^{12}√27

^{12}√4 = 4^{1/12} = ^{12}√4^{1} = ^{12}√4

Therefore, equiradical surds of the lowest order ∛2, ∜3 and ^{12}√4 are ^{12}√16, ^{12}√27and ^{12}√4 respectively.

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

^{12}√4, ∛2, ∜3.

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