COMPARISON OF SURDS

"Comparison of Surds" is the basic topic required for the students who would like to study Algebra in math. 

What is comparison of surds?

Irrational numbers of same order can be compared. If we want to compare irrational numbers with different order we have convert them with same order and then we can compare. By comparing the surds we come to which irrational number is greater or smaller.

What has to be done?

(1) Write the orders of irrational numbers given

(2) Find the least common multiple. 

(3) Make the order same.

(4) Now we can compare the radicands.

Example problems of comparision of surds:

Problem 1:

Write the irrational numbers √3,∛2,∜4 in 

(i) ascending order

(ii) descending order.

Solution:

Order of given irrational numbers are 2,3 and 4.

No we have to find the least common multiple 

L.C.M = 12

In the next step we are going to change ∛2 as 12th root 

now we are going to change the next term 4th root as 12th root 

We have changed all the given radical terms with same order.

12th root (729) is the largest number

12th root (64) is the next least number and

12th root (16) is the most least number 

Ascending order:

Ascending order means we have to write the number from least to greatest.

∛2 < ∜4 < √3  

Descending order:

Descending order means we have to write the number from greatest to least.

√3 > ∜4 > ∛2

Now let us see the next example of the topic "comparison of surds"

Problem 2:

Which is greater ∛4 or ∜5

Solution:

Order of given radical terms are 3 and 4.

To make the orders same we have to take L.C.M

L.C.M = 12

∛4 = (3 x 4) √4⁴

∛4 = 12th root(4 x 4 x 4 x 4)

      = 12th root (256)

∜5 = (4 x 3) √5³

∜5  = 12th root(5 x 5 x 5) 

      = 12th root (125)

12th root (256) > 12th root (125)

∛4 > ∜5

∛4 is greater than ∜5.

Now let us see the next example of the topic "comparison of surds"

Problem 3:

Which is greater ∛3 or √2

Solution:

Order of given radical terms are 3 and 2.

To make the orders same we have to take L.C.M

L.C.M = 6

∛3  = (3 x 2) √2²

∛3  = 6th root(2 x 2)

      = 6th root (4)

√2 = (2 x 3) √2³

√2 = 6th root(2 x 2 x 2) 

      = 6th root (8)

6th root (8) > 6th root (4)

 √2  > ∛3

√2  is greater than ∛3

Now let us see the next example of the topic "comparison of surds"

Problem 4:

Which is greater ∛3 or ∜4

Solution:

Order of given radical terms are 3 and 4.

To make the orders same we have to take L.C.M

L.C.M = 12

∛3  = (3 x 4) √3⁴

∛3  = 12th root(3 x 3 x 3 x 3)

      = 12th root (81)

∜4 = (4 x 3) √4³

∜4 = 12th root(4 x 4 x 4) 

      = 12th root (64)

12th root (81) > 12th root (64)

 ∛3   > ∜4

∜4 is greater than ∛3

Now let us see the next example of the topic "comparison of surds"

Problem 5:

Which is greater ∛3 or ∜10

Solution:

Order of given radical terms are 3 and 4.

To make the orders same we have to take L.C.M

L.C.M = 12

∛3  = (3 x 4) √3⁴

∛3  = 12th root(3 x 3 x 3 x 3)

      = 12th root (81)

∜10 = (4 x 3) √10³

∜10 = 12th root(10 x 10 x 10) 

      = 12th root (1000)

12th root (1000) > 12th root (81)

 ∜10   > ∛3

 ∜10 is greater than ∛3

Related topics of "comparison of surds"

• Simplifying radical expressions
• Radical
• Like terms and unlike terms
• Markup and markdown
• Percents grater than 1

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