Subscribe to our βΆοΈYouTube channelπ΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
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) nβa and nβb, we have nβa > nβb when x > y.
For example,
(i) β5 > β3, since 5 > 3
(ii) β21 < β28, since 21 < 28.
(iii) β10 > β6, since 10 > 6.
Comparison of surds can be done to compare the values of surd, which one has higher value and which has lower value. As the values of surds are irrational numbers and difficult to find out also as all have under root values, we have to follow some logic for comparing the values of surds.
A rational number x can be written in any order as given below.
x = 2βx2 = 3βx3 = 4βx4 = nβxn
Now if a > b, then nβa > nβb.
So from the above two points, it is clear that a surd can be expressed in different orders in the multiple of its order of minimum value and two or more surds can be compared when the surds are in same order or equiradical.
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 β7 and β5 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, β7 = 71/3 = 74/12/ = 12β74 = 12β2401 and
β5 = 51/4 = 53/12 = 12β53 = 12β125
Clearly, we see that 2401 > 125
Therefore, β7 > β5.
If the surds are not in same order or non-equiradical, then we need to express the surds in the order of Lowest Common Multiple (LCM) of other surds. By this way, all the surds can be written in same order and we can compare their values by comparing the values of radicand.
For example we need to compare the following surds and arrange them in a descending order.
2β3, 3β5,4β12.
The surds are in the order of 2, 3, and 4 respectively. If we need to compare their values, we need to express them in same order. As the LCM of 2, 3, and 4 is 12, we should express the surds in order 12. We know surds can be expressed in any order in multiple of their lowest order.
2β3 = 312 = 3612= 729112 = 12β729
3β5 = 513 = 5412= 625112 = 12β625
4β12 = 1214 = 12312 = 1728112 = 12β1728
Now all three surds are expressed in same order. As 1728 > 729 > 625
the descending order of the surds will be 4β12, 2β3, 3β5.
Examples of comparison of surds:
We will solve some similar problems to understand more on how to compare the values of surds.
1. Convert each of the following surds into equiradical surds of the lowest order and then arrange them in ascending order.
β2, β3 and 12β4
Solution:
β2, β3 and 12β4
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:
β2 = 21/3 = 24/12 = 12β24 = 12β16
β3 = 31/4 = 33/12 = 12β33 = 12β27
12β4 = 41/12 = 12β41 = 12β4
Therefore, equiradical surds of the lowest order β2, β3 and 12β4 are 12β16, 12β27 and 12β4 respectively.
Clearly, 4 < 16 < 27; hence the required ascending order of the given surds is:
12β4, β2, β3
2. Arrange the following simple surds descending order.
2β5, 3β7,6β50
Solution:
The given surds are 2β5, 3β7,6β50.
The surds are in the order of 2, 3, and 6 respectively. As the LCM of 2, 3, and 6 is 6, we should express the surds in order 6.
2β5 = 512 = 536= 12516 = 6β125
3β7 = 713 = 726 = 4916 = 6β49
6β50 need not to be changed as it is already in order 6.
As 125, 49 and 50 are the radicands of the surds in same order that is 6, the descending order of the given surds is 2β5, 6β50, 3β7.
3. Arrange the following simple surds descending order.
23β10, 32β7, 52β3
Solution:
If we need to compare the values of the given simple surds, we have to express them in the form of pure surds and after that we need to express them as equiradical surds. As the orders of the surds are 3, 2, 2 and LCM of the order is 6 we need change the order of the surds to 6.
23β10 = 3β23Γ10 = 3β8Γ10 = 3β80 = 8013 = 8026 = 640016 = 6β6400
32β7 = 2β32Γ7 = 2β9Γ7 = 2β63 = 6312 = 6336 = 25004716 = 6β250047
52β3 = 2β52Γ3 = 2β25Γ3 = 2β75 =7512 = 8036 = 51200016 = 6β512000
Hence the descending order of the given surds is 52β3, 32β7, 23β10.
11 and 12 Grade Math
From Comparison of Surds to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 17, 25 01:06 AM
Jul 17, 25 12:40 AM
Jul 16, 25 11:46 PM
Jul 16, 25 02:33 AM
Jul 15, 25 11:46 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.