SQUARE ROOT OF DECIMALS WITHOUT CALCULATOR

Find :

Problem 1:

√0.01

Solution :

= √0.01

After the decimal, we have two digits. So multiply the numerator and denominator by 100.

= √1/100

= √1/√100

= 1/10

= 0.1

Problem 2 :

√0.25

Solution :

= √0.25

After the decimal, we have two digits. So multiply the numerator and denominator by 100.

= √(25/100)

= √25/√100

= √(5⋅5)/√(10⋅10)

= 5/10

= 0.5

Problem 3 :

√0.81

Solution :

= √0.81

= √81/100

= √81/√100

= √(9⋅9)/√(10⋅10)

= 9/10

= 0.9

Problem 4 :

√0.6889

Solution :

= √0.6889

= √6889/10000

= √6889/√10000

Since the unit digit is 9 and starts with 68.

83 x 83 = 6889

87 x 87 = 7569

= 83/100

= 0.83

Problem 5 :

√0.7921

Solution :

= √0.7921

= √7921/10000

= √7921/√10000

Since the unit digit is 1 and starts with 79.

87 x 87 = 7569

89 x 89 = 7921

= 89/100

= 0.89

Problem 6 :

√0.9025

Solution :

= √0.9025

= √9025/10000

= √9025/√10000

Since the unit digit is 5 and starts with 90.

85 x 85 = 7225

95 x 95 = 9025

= 95/100

= 0.95

Problem 7 :

√48.8601

Solution :

= √48.8601

= √488601/10000

= √488601/√10000

= 699/100

= 6.99

Square Root Using Long Division

Simplify the square roots.

Problem 1 :

√35.88

Solution :

So, the square root of 35.88 is 5.989.

Problem 2 :

√36.46

Solution :

So, the square root of 36.46 is 6.038.

Problem 3 :

√89.87

Solution :

So, the square root of 89.87 is 9.479.

Problem 4 :

√99.80

Solution :

So, the square root of 99.80 is 9.989.

Problem 5 :

√62.25

Solution :

So, the square root of 62.25 is 7.889.

Problem 6 :

√97.81

Solution :

So, the square root of 97.81 is 9.889.

Problem 7 :

The cost C (in dollars) of making a square window with a side length of n inches is represented by

C = n²/5 + 175

A window costs $355. What is the length (in feet) of the window?

Solution :

Cost of square window = 355

C = n²/5 + 175

Applying C = 355

355 = n²/5 + 175

355 - 175 = n²/5

180 = n²/5

180(5) = n² 

n = √180(5)

= √900

= √30 x 30

= 30

So, the value of n is 30

Problem 8 :

The area of the triangle is represented by the formula

A = √s(s − 21)(s − 17)(s − 10)

where s is equal to half the perimeter. What is the height of the triangle

Solution :

A = √s(s − 21)(s − 17)(s − 10)

Side lengths are 17 cm, 10 cm and 21 cm

s = (a + b + c)/2

= (17 + 10 + 21) / 2

= 48/2

= 24

Applying the value of s, we get

= √24(24 − 21)(24 − 17)(24 − 10)

= √24 x 3 x 7 x 14

= √(2 x 2 x 2 x 3 x 3 x 7 x 2 x 7)

= 2 x 2 x 3 x 7

= 84 cm²

Problem 9 :

The kinetic energy K (in joules) of a falling apple is represented by K = v²/2 , where v is the speed of the apple (in meters per second). How fast is the apple traveling when the kinetic energy is 32 joules?

Solution :

K = v²/2

Here k = 32 joules

32 = v²/2

32(2) = v²

v² = 64

v = √(8 x 8)

v = 8

Problem 10 :

Copy and complete the statement with <, >, or =.

a) √81   ____  8

b) 0.5 ___ √0.25

c) 3/2 ____ √(25/4)

Solution :

a) √81   ____  8

√81 = √(9 x 9) ==> 9

9 > 8

b) 0.5 ___ √0.25

√0.25 = √0.5 x 0.5

= 0.5

0.5 = 0.5

c) 3/2 ____ √(25/4)

√(25/4) = √(5 x 5) / (2 x 2)

= 5/2

= 2.5

3/2 = 1.5

1.5 < 2.5

Problem 11 :

The area of a sail is 40 1/2 square feet. The base and the height of the sail are equal. What is the height of the sail (in feet)?

Solution :

Area of sail = 40  1/2

Base = b and height = h

(1/2) ⋅ b ⋅ h = 81/2

b ⋅ h = (81/2) (2)

Here b and h are equal, then 

b² = 81

b = 9

So, base and height of the triangle is 9 feet.

Problem 12 :

Evaluate 

a) 3√16 - 5

b)  10 - 4√(1/16)

c) √6.76 + 5.4

d) 8√8.41 + 1.8

Solution :

a) 3√16 - 5

= 3√(4 x 4) - 5

= 3(4) - 5

= 12 - 5

= 7

b)  10 - 4√(1/16)

= 10 - 4√(1 x 1) / (4 x 4)

= 10 - 4(1/4)

= 10 - 1

= 9

c) √6.76 + 5.4

= √(676/100) + 5.4

= √(26 x 26) / (10 x 10) + 5.4

= 26/10 + 5.4

= 2.6 + 5.4

= 8

d) 8√8.41 + 1.8

= 8√(841/100) + 1.8

= 8√(29 x 29)/(10 x 10) + 1.8

= 8 (2.9/10) + 1.8

= 23.2/10 + 1.8

= 2.32 + 1.8

= 4.12