Loading [MathJax]/jax/output/HTML-CSS/jax.js

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) n√a and n√b, we have n√a > n√b when x > y.

For example,

(i) √5 > √3, since 5 > 3

(ii) βˆ›21 < βˆ›28, since 21 < 28.

(iii) ∜10 > ∜6, 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 = 2√x2 = 3√x3 = 4√x4 = n√xn

Now if a > b, then n√a > n√b.

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 βˆ›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, βˆ›7 = 71/3 = 74/12/ = 12√74 = 12√2401 and

∜5 = 51/4 = 53/12 = 12√53 = 12√125

Clearly, we see that 2401 > 125

Therefore, βˆ›7 > ∜5.

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.

2√3, 3√5,4√12.

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.

2√3 = 312 = 3612= 729112 = 12√729

3√5 = 513 = 5412= 625112 = 12√625

4√12 = 1214 = 12312 = 1728112 = 12√1728

Now all three surds are expressed in same order. As 1728 > 729 > 625 

the descending order of the surds will be 4√12, 2√3, 3√5.



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.

                                   βˆ›2, ∜3 and 12√4

Solution:

βˆ›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:

βˆ›2 = 21/3 = 24/12 = 12√24 = 12√16

∜3 = 31/4 = 33/12 = 12√33 = 12√27

12√4 = 41/12 = 12√41 = 12√4

Therefore, equiradical surds of the lowest order βˆ›2, ∜3 and 12√4 are 12√16, 12√27 and 12√4 respectively.

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

12√4, βˆ›2, ∜3


2. Arrange the following simple surds descending order.

2√5, 3√7,6√50

Solution:

The given surds are 2√5, 3√7,6√50.

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.

2√5 = 512 = 536= 12516 = 6√125

3√7 = 713 = 726 = 4916 = 6√49

6√50 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 2√5, 6√50, 3√7.


3. Arrange the following simple surds descending order.

23√10, 32√7, 52√3

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.

23√10 = 3√23Γ—10 = 3√8Γ—10 = 3√80 = 8013 = 8026 = 640016 = 6√6400

32√7 = 2√32Γ—7 = 2√9Γ—7 = 2√63 = 6312 = 6336 = 25004716 = 6√250047

52√3 = 2√52Γ—3 = 2√25Γ—3 = 2√75 =7512 = 8036 = 51200016 = 6√512000

Hence the descending order of the given surds is 52√3, 32√7, 23√10.






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.



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.




Share this page: What’s this?

Recent Articles

  1. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  2. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  3. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  4. Perimeter of a Figure | Perimeter of a Simple Closed Figure | Examples

    Jul 16, 25 02:33 AM

    Perimeter of a Figure
    Perimeter of a figure is explained here. Perimeter is the total length of the boundary of a closed figure. The perimeter of a simple closed figure is the sum of the measures of line-segments which hav…

    Read More

  5. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More