COMPARISON OF SURDS

In this section, you will learn how to compare equiradical surds and non-equiradical surds. 

Equiradical Surds : 

Surds of the same order 

Example : 

√3, √6, √8

Non-Equiradical Surds : 

Surds of different orders 

Example : 

√3, ³√6, ⁴√8

Radicand : 

The value inside the radical sign.

Example : 

In √3, 3 is the radicand.

Comparing Equiradical Surds

To compare two or more equiradical surds, we have to compare the radicands.

For example, let us consider the following two equiradical surds. 

³√7 and ³√5

In the above two equiradical surds, the radicands are 7 and 5.

And also, 

7  >  5

Then

³√7  >  ³√5

Comparing Non-Equi Radical Surds

To compare two or more non-equiradical surds, first we have to convert them into equiradical surds. Then, we have compare the radicands of the equiradical surds.

Using the least common multiple of the orders of non-equiradical surds, we can convert them into equiradical surds.  

For example, let us consider the following non-equiradical surds. 

√3, ³√2 and ⁴√4

The orders of the above surds are 2, 3 and 4. 

Thee least common multiple of (2, 3 and 4) is 12.

So, we have to make the order of each surd as 12. 

Now, all the surds are expressed in the same order. 

Compare the radicands :

729 > 64 > 16

Then, 

¹²√729 > ¹²√64 > ¹²√16

Therefore, 

√3 > ⁴√4 > ³√2

Comparing Surds with Same Radicand

If two surds of different orders have the same radicand, then the surd with the smaller order will be greater in value.  

For example, 

³√5 > ⁵√5

Solved Problems

Problem 1 :

Which is greater ?

√4 or √6

Solution :

The above two surds have the same  order (i.e., 2).

To compare the above surds, we have to compare the radicands 4 and 6.

Clearly 6 is greater than 4.

So, √6 is greater than √4.

√6 > √4

Problem 2 :

Which is greater ?

√2 or ³√3

Solution :

The above two surds have different orders. The are 2 and 3.

Using the least common multiple of the orders 2 and 3, we can convert them into surds of same order.

Least common multiple of (2 and 3) is 6.

√2 = ²√2 = ^3x2√(2³) = ⁶√8

³√3 = ^2x3√(3²) = ⁶√9

Now, the given two surds are expressed in the same order.

Compare the radicands :

9 > 8

Then,

⁶√9 > ⁶√8

Therefore,

³√3 > √2

Problem 3 :

Which is greater ?

⁴√3 or ⁶√4

Solution :

The above two surds have different orders. The are 4 and 6.

Using the least common multiple of the orders 4 and 6, we can convert them into surds of same order.

Least common multiple of (4 and 6) is 12.

⁴√3 = ^4x3√(3³) = ¹²√27

⁶√4 = ^6x2√(4²) = ¹²√16

Now, the given two surds are expressed in the same order.

Compare the radicands :

27 > 16

Then,

¹²√27 > ¹²√16

Therefore,

⁴√3 > ⁶√4

Problem 4 :

Which is greater ?

⁴√4 or ⁵√5

Solution :

The above two surds have different orders. The are 4 and 5.

Using the least common multiple of the orders 4 and 5, we can convert them into surds of same order.

Least common multiple of (4 and 5) is 20.

Then,

⁴√4 = ^4x5√(4⁵) = ²⁰√1024

⁵√5 = ^5x4√(5⁴) = ²⁰√625

Now, the given two surds are expressed in the same order.

Compare the radicands :

1024 > 625

Then,

²⁰√1024 > ²⁰√625

Therefore,

⁴√4 > ⁵√5

Problem 5 :

Which is greater ?

⁷√25 or ⁵√25

Solution :

The above two surds have different orders with the same radicand.

Then, the surd with the smaller order will be greater in value.

Therefore, ⁵√25 is greater than ⁷√25.

That is,

⁵√25 > ⁷√25

Problem 6 :

Arrange the following surds in ascending order :

³√4, ⁶√5  and ⁴√6

Solution :

The orders of the above surds are 3, 6 and 4.

The least common multiple of (3, 6 and 4) is 12.

So, we have to make the order of each surd as 12.

Then,

³√4 = ^3x4√(4⁴) = ¹²√256

⁶√5 = ^6x2√(5²) = ¹²√25

⁴√6 = ^4x3√(6³) = ¹²√216

Now, the given two surds are expressed in the same order.

Arrange the radicands in ascending order :

25, 216, 256

Then,

¹²√25, ¹²√216, ¹²√256

Therefore, the ascending order of the given surds is

⁶√5, ⁴√6, ³√4

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