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## Note on Square Root

• Note
• Things to remember
• Exercise
• Quiz

The process by which we can make opposite of square is called finding the square root. It is the opposite of squaring. It is a number that when multiplied by itself an indicated number of times forms a product equal to a specified number.

For example:

• The square root of 9 is 3 (because 3² = 3x3 = 9)
• The square root of 16 is 4 (because 42 = 4x4 =16)
• The square root of 36 is 6 (because 6² = 6x6 = 36)
• The square root of 81 is 9 (because 92 = 9x9 = 81)
• The square root of 100 is 10 (because 10² = 10x10 = 100)

$$\sqrt{}$$ represent the square root.

$$\sqrt{25}$$ means the square root of 25

$$\sqrt{64}$$ means square root of 64

A natural number are the perfect square root. Some of the natural numbers are 1, 4, 9, 16, etc.

For example: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 etc. are perfect squares.

A square root of a number can be done by two methods which make easy in the process of factorization. The two methods are:

1. Prime factorization method
2. Division method

Prime factorization method make easy in finding out the square root in the natural number.

• The number should be the factor of the prime number or should be expressed as the factor of a prime number.
• Make pairs of the factor and each pair should be equal.
• Take one factor from each pair.
• The product of the taken factor is the square root of the given number.

It can be shown by the numerical examples:

1. Find the square root of 36.

Solution:

$$\sqrt{36}$$

= $$\sqrt{6 \times 6}$$

= 6

2. Find the square root of 2025.

Solution:

$$\sqrt{2025}$$

= $$\sqrt{9\times 9\times 5\times 5}$$

= 9×5

= 45

 5 2025 5 405 9 81 9

### Square root by the division method

Division method is a faster way to find out the square root of a number. It is less time consuming then the factorization method. For example, the number 512490 is grouped into three pairs of 51, 24, 90. If the number of digits in the number is odd then the first group will have one digit and rest will have two digits. For example, the number 18021 is grouped into three groups of 1, 80, 21. Cube Root.

In this method, we make the pair of digit whose square root has to be found. While pairing the digit we do it from the right side. So that if the number of digits is even then all group will have 2 digits and if a number of digits are odd then the first group has one and other will have two digits. For example, the number 512490 is grouped into three pairs of 51, 24, 90. If the number of digits in the number is odd then the first group will have one digit and rest will have two digits. For example, the number 18021 is grouped into three groups of 1, 80, 21. Cube Root.

Examples

1. Find the square root of 441

Solution:

$$\sqrt{441}$$

= $$\sqrt{3×3×7×7}$$

= 3× 7

= 21

2. Simplify: $$\sqrt{4^2×2^2}$$

Solution:

$$\sqrt{4^2×2^2}$$

= 4×2

= 8

•  The square root of the number which is multiplied by itself gives you the original number.
• Its symbol is called a radical number and looks like this $$\sqrt{4}$$ is 2, because 2x2 = 4.
• The second root is usually called the Cube.
• The opposite of squaring a number is called finding Square Root.
Eg.;The square root of 25 is 5 (because 25 = 5x5 =25 ).

.

### Very Short Questions

Solution:

Square of 4

= 42

= 4×4

= 16

Solution:

Square of 10

= 102

= 10×10

= 100

Solution:

$$\sqrt{1+3}$$

= $$\sqrt{4}$$

= $$\sqrt{2×2}$$

= 2

Solution:

$$\sqrt{1+3+5}$$

= $$\sqrt{9}$$

=$$\sqrt{3×3}$$

= 3

Solution:

$$\sqrt{1+3+5+7.....+99}$$

= $$\frac{99+1}{2}$$

= $$\frac{100}{2}$$

= 50

Solution:

$$\sqrt{1+3+5+7}$$

= $$\sqrt{16}$$

=$$\sqrt{4×4}$$

= 4

Solution:

(0.2)2

= (0.2)×(0.2)

= 0.04

Solution:

(0.01)2

= (0.01)×(0.01)

=0.0001

Solution:

(-3)2

= (-3)×(-3)

=9

Solution:

Square root of 81

=$$\sqrt{81}$$

=$$\sqrt{9×9}$$

=9

Solution:

square root of 256

= $$\sqrt{256}$$

= $$\sqrt{2 × 2 ×2 ×2 ×2× 2 ×2× 2}$$

= $$\sqrt{2^2× 2^2 ×2^2 ×2^2}$$

= 2 × 2 × 2 ×2

= 16

Solution:

square root of 676

=$$\sqrt{676}$$

= $$\sqrt{2×2×13×13}$$

= 2× 13

= 26

Solution:
$$\sqrt{2^2×3^2}$$

= 2 × 3

= 6

Solution:

$$\sqrt{12}$$

= $$\sqrt{2×2×3}$$

= 2$$\sqrt{3}$$

Solution:

= $$\sqrt{25x^6 ×4y^2}$$

= $$\sqrt{5×5×(x^3)^2×2×2×y^2}$$

= 5× x3× 2× y

= 10x3y

0%

12
15
16
10

25
20
35
36

42
52
59
64

270
245
220
256

122
121
123
124
• ### Find the squares of: (frac{1}{2})

(frac{1}{16})
(frac{1}{8})
(frac{1}{6})
(frac{1}{4})
• ### Find the squares of: (frac{3}{4})

(frac{8}{16})
(frac{8}{15})
(frac{9}{15})
(frac{9}{16})
• ### Find the squares of: (frac{2}{3})

(frac{4}{12})
(frac{4}{9})
(frac{4}{10})
(frac{4}{14})
• ### Find the squares of: (frac{7}{10})

(frac{49}{100})
(frac{42}{100})
(frac{49}{110})
(frac{49}{90})

(frac{7}{6})
(frac{6}{8})
(frac{7}{2})
(frac{7}{8})
• ### Find the square root of: (frac{121}{256})

(frac{11}{16})
(frac{11}{18})
(frac{10}{16})
(frac{11}{20})
• ### Find the square root of:(frac{324}{625})

(frac{15}{25})
(frac{16}{22})
(frac{20}{25})
(frac{18}{25})
• ### Find the square root of: (frac{144}{400})

(frac{12}{20})
(frac{10}{20})
(frac{12}{22})
(frac{12}{26})

87
85
37
25

25
64
72
46
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