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.
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
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.
√3 = ^{2}√3
= ^{6x}^{2}√3^{6}
= ^{12}√729
^{3}√2 = ^{4x3}√2^{4}
= ^{12}√16
^{4}√4 = ^{3x4}√4^{3}
= ^{12}√64
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
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
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}√(2^{3}) = ^{6}√8
^{3}√3 = ^{2x3}√(3^{2}) = ^{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}√(3^{3}) = ^{12}√27
^{6}√4 = ^{6x2}√(4^{2}) = ^{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}√(4^{5}) = ^{20}√1024
^{5}√5 = ^{5x4}√(5^{4}) = ^{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}√(4^{4}) = ^{12}√256
^{6}√5 = ^{6x2}√(5^{2}) = ^{12}√25
^{4}√6 = ^{4x3}√(6^{3}) = ^{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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