COMPARISON OF SURDS 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 surds in ascending order :

³√4, ⁶√5  and ⁴√6

1. Solution :

√4 or √6

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

2. Solution :

√2 or ³√3

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√(2³) = ⁶√8

³√3 = ^2x3√(3²) = 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

3. Solution :

⁴√3 or ⁶√4

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√(3³) = ¹²√27

⁶√4 = ^6x2√(4²) = 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

4. Solution :

⁴√4 or ⁵√5

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√(4⁵) = ²⁰√1024

⁵√5 = ^5x4√(5⁴) = 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

5. Solution :

⁷√25 or ⁵√25

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

6. Solution :

³√4, ⁶√5  and ⁴√6

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√(4⁴) = ¹²√256

⁶√5 = ^6x2√(5²) = ¹²√25

⁴√6 = ^4x3√(6³) = 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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