SQUARE ROOTS OF FRACTIONS AND DECIMALS

Example 1 : 

Find the square root of 4/9. 

Solution :

√(4/9)  =  √4 / √9

When we decompose 4 and 9 into prime factors, we get

4  =  2²

9  =  3²

Then, we have

√(4/9)  =  √2² / √3²

√(4/9)  =  2/3

So, the square root of 4/9 is 2/3.

Example 2 : 

Find the square root of 169/289. 

Solution :

√(169)  =  √169 / √289

When we decompose 169 and 289 into prime factors, we get

169  =  13²

289  =  17²

Then, we have

√(169/289)  =  √13² / √17²

√(169/289)  =  13/17

So, the square root of 169/289 is 13/17.

Example 3 : 

Find the square root of 2.56.

Solution :

Convert the decimal number 2.56 into a fraction.

Because there are two digits after the decimal point, multiply and divide 2.56 by 100 to get rid of the decimal point. 

2.56  =  (2.56 ⋅ 100) / 100

2.56  =  256 / 100

Take square root on each side. 

√2.56  =  √(256/100)

Take square separately for numerator and denominator. 

√2.56  =  √256 / √100 -----(1)

To find the square root of 256 and 100, decompose 256 and 100 into prime factors using synthetic division as shown below.

From the above synthetic division, we have

256  =  2⁸

100  =  2² ⋅ 5²

Then, we have

(1)----->  √2.56  =  √2⁸ / √(2² ⋅ 5²)

√2.56  =  2⁴ / (2 ⋅ 5)

√2.56  =  16 / 10

√2.56  =  1.6

So, the square root of 2.56 is 1.6.

Example 4 :

Find the square root of 51.84.

Solution :

Convert the decimal number 51.84 into a fraction.

Because there are two digits after the decimal point, multiply and divide 51.84 by 100 to get rid of the decimal point. 

51.84  =  (51.84 ⋅ 100) / 100

51.84  =  5184 / 100

Take square root on each side. 

√51.84  =  √(5184/100)

Take square separately for numerator and denominator. 

√51.84  =  √5184 / √100 -----(1)

To find the square root of 5184 and 100, decompose 5184 and 100 into prime factors using synthetic division as shown below.

From the above synthetic division, we have

5184  =  2⁶ ⋅ 3⁴

100  =  2² ⋅ 5²

Then, we have

(1)----->  √51.84  =  √(2⁶ ⋅ 3⁴) / √(2² ⋅ 5²)

√51.84  =  (2³ ⋅ 3²) / (2 ⋅ 5)

√51.84  =  (8 ⋅ 9) / 10

√51.84  =  72/10

√51.84  =  7.2

So, the square root of 51.84 is 7.2.

Example 5 :

Find the square root of 0.2916.

Solution :

Convert the decimal number 0.2916 into a fraction.

Because there are four digits after the decimal point, multiply and divide 0.2916 by 10000 to get rid of the decimal point. 

0.2916  =  (0.2916 ⋅ 10000) / 10000

0.2916  =  2916 / 10000

Take square root on each side. 

√0.2916  =  √(2916/10000)

Take square separately for numerator and denominator. 

√0.2916  =  √2916 / √10000 -----(1)

To find the square root of 2916 and 10000, decompose 2916 and 10000 into prime factors using synthetic division as shown below.

From the above synthetic division, we have

2916  =  2² ⋅ 3⁶

10000  =  5⁴ ⋅ 2⁴

Then, we have

(1)----->  √0.2916  =  √(2² ⋅ 3⁶) / √(5⁴ ⋅ 2⁴)

√0.2916  =  √(2² ⋅ 3⁶) / √(5⁴ ⋅ 2⁴)

√0.2916  =  (2 ⋅ 3³) / (5² ⋅ 2²)

√0.2916  =  (2 ⋅ 27) / (25 ⋅ 4)

√0.2916  =  54/100

√0.2916  =  0.54

So, the square root of 0.2916 is 0.54.

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