ORDERING SQUARE ROOTS FROM LEAST TO GREATEST

Example 1 : 

Write the following numbers in the order from least to greatest. 

9, √63, 12, √124, √42

Solution : 

In the above numbers 9 and 12 do not have square roots.

Write 9 and 12 inside the square root as shown below.

9  =  √9²  =  √81

12  =  √12²  =  √144

Then, the given numbers are

√81, √63, √144, √124, √42

There is square root for every number above. 

Now, compare the numbers inside the square roots and write them from least to greatest. 

So,

√42, √63, √81, √124, √144

√42, √63, 9, √124, 12

Example 2 : 

Write the following numbers in the order from least to greatest. 

7, 2√3, 13, 3√6, √55

Solution : 

In the above numbers 7 and 13 do not have square roots.

Write 7 and 13 inside the square root as shown below.

7  =  √7²  =  √49

12  =  √12²  =  √144

In 2√3 and 3√6, we don't have the complete square root. 

Get complete square root for 2√3 and 3√6 as shown below.

2√3  =  (√2²)√3  =  √4√3  =  √12

3√6  =  (√3²)√6  =  √9√6  =  √54

Then, the given numbers are

√49, √12, √169, √54, √55

There is square root for every number above. 

Now, compare the numbers inside the square roots and write them from least to greatest. 

So, 

√12, √49, √54, √55, √169

2√3, 7, 3√6, √55, 13

Example 3 : 

Write the following numbers in the order from least to greatest. 

(3√5)², (√3)³, 5√2, √12

Solution : 

(3√5)² :

(3√5)²  =  (3√5)(3√5)

(3√5)²  =  (3 ⋅ 3)√(5 ⋅ 5)

(3√5)²  =  9√25

(3√5)²  =  (√9²)(√25)

(3√5)²  =  √81√25

(3√5)²  =  √2025

(√3)³ :

(√3)³  =  √3√3√3

(√3)³  =  √(3 ⋅ 3 ⋅ 3)

(√3)³  =  √27

5√2 : 

5√2  =  (√5²)√2

5√2  =  √25√2

5√2  =  √50 

Then, the given numbers are

√2025, √27, √50

There is square root for every number above. 

Now, compare the numbers inside the square roots and write them from least to greatest. 

So, 

√27, √50, √2025

(√3)³, 5√2, (3√5)²

Example 4 : 

Write the following numbers in the order from least to greatest. 

(√64 + √196), (√40 + √160), (√17√51)

Solution : 

√64 + √196 : 

√64 + √196  =  8 + 14

√64 + √196  =  22

√64 + √196  =  √22²

√64 + √196  =  √484

√40 + √160 :

√40 + √160  =  2√10 + 4√10

√40 + √160  =  6√10

√40 + √160  =  (√6²)√10

√40 + √160  =  √36√10

√40 + √160  =  √360

√17√51 : 

√17√51  =  √(17 ⋅ 51)

√17√51  =  √867

Then, the given numbers are

√484, √360, √867

There is square root for every number above. 

Now, compare the numbers inside the square roots and write them from least to greatest. 

So, 

√360, √484, √867

(√40 + √160), (√64 + √196), (√17√51)

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