COMPARING RADICALS WORKSHEET

Problem 1 :

Which is greater  ? 

√4 or √6

Problem 2 :

Which is greater ? 

√2 or ³√3

Problem 3 :

Which is greater ? 

⁴√3 or ⁶√4

Problem 4 :

Which is greater ? 

⁴√4 or ⁵√5

Problem 5 :

Which is greater ? 

⁷√25 or ⁵√25

Problem 6 :

Arrange the following radicals in ascending order :

³√4, ⁶√5  and ⁴√6

Solutions

Problem 1 :

Which is greater  ? 

√4 or √6

Solution:

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

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

Clearly 6 is greater than 4. 

So, √6 is greater than √4. 

That is, 

√6  >  √4

Problem 2 :

Which is greater ? 

√2 or ³√3

Solution:

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

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

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

Then, 

√2  =  ^2x3√(2³)  =  ⁶√8

³√3  =  ^3x2√(3²)  =  ⁶√9

Now, the given two radicals 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 radicals have different orders. The are 4 and 6.

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

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

Then, 

⁴√3  =  ^4x3√(3³)  =  ¹²√27

⁶√4  =  ^6x2√(4²)  =  ¹²√16

Now, the given two radicals 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 radicals 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 radicals 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 radicals have different orders with the same radicand.

Then, the radical 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 radicals in ascending order :

³√4, ⁶√5  and ⁴√6

Solution:

The orders of the above radicals 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 radical 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 radicals 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 radicals is 

⁶√5, ⁴√6, ³√4

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