Subject: Compulsory Maths

The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number.

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 4
^{2}= 4x4 =16) - The square root of 36 is 6 (because 6² = 6x6 = 36)
- The square root of 81 is 9 (because 9
^{2}= 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:

- Prime factorization method
- 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 |

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^{2 }= 5x5 =25 ).

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Find the squares roots of 4

Solution:

Square of 4

= 4^{2}

= 4×4

= 16

Find the squares roots of 10

Solution:

Square of 10

= 10^{2}

= 10×10

= 100

Solve: \(\sqrt{1+3}\)

Solution:

\(\sqrt{1+3}\)

= \(\sqrt{4}\)

= \(\sqrt{2×2}\)

= 2

Solve: \(\sqrt{1+3+5}\)

Solution:

\(\sqrt{1+3+5}\)

= \(\sqrt{9}\)

=\(\sqrt{3×3}\)

= 3

Solve: \(\sqrt{1+3+5+7.....+99}\)

Solution:

\(\sqrt{1+3+5+7.....+99}\)

= \(\frac{99+1}{2}\)

= \(\frac{100}{2}\)

= 50

Solve: \(\sqrt{1+3+5+7}\)

Solution:

\(\sqrt{1+3+5+7}\)

= \(\sqrt{16}\)

=\(\sqrt{4×4}\)

= 4

Solve: (0.2)^{2}

Solution:

(0.2)^{2}

= (0.2)×(0.2)

= 0.04

Solve: (0.01)^{2}

Solution:

(0.01)^{2}

= (0.01)×(0.01)

=0.0001

Solve: (-3)^{2}

Solution:

(-3)^{2}

= (-3)×(-3)

=9

Solve: Square root of 81

Solution:

Square root of 81

=\(\sqrt{81}\)

=\(\sqrt{9×9}\)

=9

Find the square root of 256

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

Find the square root of 676

Solution:

square root of 676

=\(\sqrt{676}\)

= \(\sqrt{2×2×13×13}\)

= 2× 13

= 26

Simplify: \(\sqrt{2^2×3^2}\)

Solution:

\(\sqrt{2^2×3^2}\)

= 2 × 3

= 6

Simplify: \(\sqrt{12}\)

Solution:

\(\sqrt{12}\)

= \(\sqrt{2×2×3}\)

= 2\(\sqrt{3}\)

Simplify: \(\sqrt{25x^6 ×4y^2}\)

Solution:

= \(\sqrt{25x^6 ×4y^2}\)

= \(\sqrt{5×5×(x^3)^2×2×2×y^2}\)

= 5^{}× x^{3}× 2^{}× y

= 10x^{3}y

Find the square root of \(\frac{144}{400}\)

Solution:

square root of \(\frac{144}{400}\)

= \(\sqrt\frac{144}{400}\)

= \(\sqrt\frac{12×12}{20×20}\)

= \(\frac{12}{20}\)

=\(\frac{3}{5}\)

Find the square root of \(\frac{ 324}{625}\)

Solution:

square root of \(\frac{324}{625}\)

= \(\sqrt\frac{324}{625}\)

= \(\sqrt\frac{18×18}{25×25}\)

= \(\sqrt\frac{18}{25}\)

Find the square root of 1764.

Solution:

2 | 1764 | |

2 | 882 | |

3 | 441 | |

3 | 147 | |

7 | 49 | |

7 |

\(\sqrt{1764}\)

= \(\sqrt{2×2×3×3×7×7}\)

= 2× 3× 7

= 42

Find the square root of 576.

Solution:

2 | 576 |

2 | 288 |

2 | 144 |

2 | 72 |

2 | 36 |

2 | 18 |

2 | 9 |

3 |

\(\sqrt{576}\)

= \(\sqrt{2×2×2×2×2×2×3×3}\)

= 2×2×2×3

= 24

Find the square root of \(\frac{64}{144}\)

Solution:

square root of \(\frac{64}{144}\)

= \(\sqrt{\frac{64}{144}}\)

= \(\sqrt{\frac{2×2×2×2×2×2}{2×2×2×2×3×3}}\)

= \(\sqrt{\frac{2×2}{3×3}}\)

= \(\frac{2}{3}\)

Find the square root of \(\frac{441}{1764}\)

Solution:

square root of \(\frac{441}{1764}\)

= \(\sqrt{\frac{441}{1764}}\)

= \(\sqrt{\frac{3×3×7×7}{2×2×3×3×7×7}}\)

= \(\sqrt{\frac{1}{2×2}}\)

= \(\sqrt{\frac{1^2}{2^2}}\)

= \(\frac{1}{2}\)

Find the square root of \(\frac{400}{441}\).

Solution:

Square root of \(\frac{400}{441}\)

= \(\sqrt{\frac{400}{441}}\)

= \(\sqrt{\frac{2×2×2×2×5×5}{3×3×7×7}}\)

= \(\sqrt{\frac{2^2×2^2×5^2}{3^2×7^2}}\)

= \(\frac{2×2×5}{3×7}\)

= \(\frac{20}{21}\)

Find the smallest number by which the number must be divided to obtain a perfect square.

512

Solution:

512

2 | 512 |

2 | 256 |

2 | 128 |

2 | 64 |

2 | 32 |

2 | 16 |

2 | 8 |

2 | 4 |

2 |

512 = 2^{2}× 2^{2}×2^{2}×2^{2}×2

∴ The required number is 2.

Find the square root of 6561.

Solution:

Square root of 6561

= \(\sqrt{6561}\)

= \(\sqrt{3×3×3×3×3×3×3×3}\)

= \(\sqrt{3^2× 3^2×3^2×3^2}\)

= 3 × 3 × 3 × 3

= 81

Find the square root of 4096.

Solution:

Square root of 4096

= \(\sqrt{4096}\)

= \(\sqrt{2×2×2×2×2×2×2×2×2×2×2×2}\)

= \(\sqrt{2^2×2^2×2^2×2^2×2^2×2^2}\)

= 2×2×2×2×2×2

= 64

∴ The square root of 4096 = 64.

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