## Table of Contents

Welcome to the world of square roots! In this article, we will learn **how to find the square root of a number** by 4 methods, Prime Factorization, Repeated Subtraction, Long Division, and Estimation Method.

## What is Square Root?

The square root is a mathematical operation that helps us find the value which, when multiplied by itself, gives a specified number. In other words, it is the value that, when squared, equals the original number. The square root of a number is denoted by the symbol √.

### Understanding the Concept of Square Root

Imagine you have a number, let’s say 25. The square root of 25 is a value that, when multiplied by itself, gives the original number. In this case, the square root of 25 is 5, since 5 multiplied by 5 equals 25. Square roots help us find the side length of a square when we know its area or determine the original number from its square.

### The Significance of Square Roots

Square roots have a wide range of applications in various fields such as geometry, mensuration, and physics. They are essential for solving equations, working with quadratic functions, and understanding geometric shapes. By mastering the skill of finding square roots, you’ll enhance your problem-solving abilities and unlock a deeper understanding of mathematical concepts.

## Challenges Faced by Students in Finding Square Roots

### Complex calculations

Calculating square roots often involves performing complex calculations, which can be time-consuming and prone to errors. Students may struggle with the computational aspect, especially when dealing with larger numbers or decimals.

### Limited knowledge of different methods

There are multiple methods available for finding square roots, such as prime factorization, repeated subtraction, long division, and estimation. Students may not be aware of these methods or may not have enough practice in using them effectively.

### Difficulties in Identifying Perfect and Non-Perfect Squares

Differentiating between perfect squares (numbers that have whole number square roots) and non-perfect squares can be a challenge. We’ll provide you with techniques to quickly identify them and simplify the process.

We will learn to solve all these problems.

## How to Find Square Root of Numbers?

Finding the square root of numbers can be approached using various methods. Let’s explore some commonly used techniques:

### Find the Square Root by Prime Factorization

This method involves breaking down the given number into its prime factors and grouping them in pairs. The square root is then determined by taking one factor from each pair. By simplifying the prime factors, we can find the square root of the original number.

Here are five examples of finding the **square root of numbers using the prime factorization method**:

Number | Prime Factorization | Single digit from Pair | Square Root |
---|---|---|---|

16 | 2×2×2×2 | 2×2 | √16 = 2×2 = 4 |

81 | 3×3×3×3 | 3×3 | √81 = 3×3 = 9 |

100 | 2×2×5×5 | 2×5 | √100 = 2×5 = 10 |

225 | 3×3×5×5 | 3×5 | √225 = 3×5 = 15 |

400 | 2×2×2×2×5×5 | 2×2×5 | √400 = 2×2×5 = 20 |

**Example 1**: Find the square root of 16.

Step 1: Prime factorize the number 16. 16 = 2^{4} = (2 x 2 x 2 x 2)

Step 2: Group the prime factors in pairs. (2^2) = (2 x 2 x 2 x 2) = (2 x 2) (2 x 2) = 2^{2}

Step 3: Take the multiplication from the group = 2^{2} = 4

Therefore, the square root of 16 is 4.

**Example 2**: Find the square root of 81.

Step 1: Prime factorize the number 81. 81 = 3^{4} = (3 x 3 x 3 x 3)

Step 2: Group the prime factors in pairs. (3^2) = (3 x 3 x 3 x 3) = (3 x 3) (3 x 3) = 3^{2}

Step 3: Take the multiplication from the group = 3^{2} = 9

Therefore, the square root of 81 is 9.

**Example 3**: Find the square root of 100.

Step 1: Prime factorize the number 100. 100 = 2^{2} x 5^{2}

Step 2: Group the prime factors in pairs. (2^1 x 5^1)

Step 3: Take the multiplication from the group. 2 x 5 = 10

Therefore, the square root of 100 is 10.

**Example 4**: Find the square root of 225.

Step 1: Prime factorize the number 225. 225 = 3^{2} x 5^{2}

Step 2: Group the prime factors in pairs. (3^1 x 5^1)

Step 3: Take the multiplication from the group. 3 x 5 = 15

Therefore, the square root of 225 is 15.

**Example 5**: Find the square root of 400.

Step 1: Prime factorize the number 400. 400 = 2^{4} x 5^{2} = (2 x 2 x 2 x 2 x 5 x 5)

Step 2: Group the prime factors in pairs. (2^{2} x 2^{2} x 5^{2})

Step 3: Take the multiplication from the group . 2 x 2 x 5 = 20

Therefore, the square root of 400 is 20.

find the square root of 1764 by prime factorization method

By using the prime factorization method, we can easily find the square root of various numbers by identifying their prime factors, grouping them in pairs, and taking one factor from each pair to calculate the square root.

### Find the Square Root by Repeated Subtraction Method

In this method, we start with a number and repeatedly subtract the consecutive odd numbers from it until we reach zero. The count of subtractions performed gives us the square root of the original number. This method is particularly useful for finding square roots of smaller numbers.

Certainly! Here are five examples of finding the square root of numbers using the repeated subtraction method:

**Example 1**: Find the square root of 36 using the repeated subtraction method.

Starting with 36, subtract successive odd numbers until reaching zero. The number of subtractions performed is 6, so the square root of 36 is 6.

Number | Sequence |
---|---|

36 – 1 = 35 | 1 |

35 – 3 = 32 | 2 |

32 – 5 = 27 | 3 |

27 – 7 = 20 | 4 |

20 – 9 = 11 | 5 |

11 – 11 = 0 | 6 |

**Example 2**: Find the square root of 64 using the repeated subtraction method.

Starting with 64, subtract successive odd numbers until reaching zero. The number of subtractions performed is 8, so the square root of 64 is 8.

Number | Sequence |
---|---|

64 – 1 = 63 | 1 |

63 – 3 = 60 | 2 |

60 – 5 = 55 | 3 |

55 – 7 = 48 | 4 |

48 – 9 = 39 | 5 |

39 – 11 = 28 | 6 |

28 – 13 = 15 | 7 |

15 – 15 = 0 | 8 |

**Example 3**: Find the square root of 100 by the method of repeated subtraction

Starting with 100, subtract successive odd numbers until reaching zero. The number of subtractions performed is 10, so the square root of 100 is 10.

Number | Sequence |
---|---|

100 – 1 = 99 | 1 |

99 – 3 = 96 | 2 |

96 – 5 = 91 | 3 |

91 – 7 = 84 | 4 |

84 – 9 = 75 | 5 |

75 – 11 = 64 | 6 |

64 – 13 = 51 | 7 |

51 – 15 = 36 | 8 |

36 – 17 = 19 | 9 |

19 – 19 = 0 | 10 |

**Example 4**: Find the square root of 144 by the method of repeated subtraction

Starting with 144, subtract successive odd numbers until reaching zero. The number of subtractions performed is 12, so the square root of 144 is 12.

Number | Sequence |
---|---|

144 – 1 = 143 | 1 |

143 – 3 = 140 | 2 |

140 – 5 = 135 | 3 |

135 – 7 = 128 | 4 |

128 – 9 = 119 | 5 |

119 – 11 = 108 | 6 |

108 – 13 = 95 | 7 |

95 – 15 = 80 | 8 |

80 – 17 = 63 | 9 |

63 – 19 = 44 | 10 |

44 – 21 = 23 | 11 |

23 – 23 = 0 | 12 |

**Example 5**: Find the square root of 225 using the repeated subtraction method.

Starting with 225, subtract successive odd numbers until reaching zero. The number of subtractions performed is 15, so the square root of 225 is 15.

Number | Sequence |
---|---|

225 – 1 = 224 | 1 |

224 – 3 = 221 | 2 |

221 – 5 = 216 | 3 |

216 – 7 = 209 | 4 |

209 – 9 = 200 | 5 |

200 – 11 = 189 | 6 |

189 – 13 = 176 | 7 |

176 – 15 = 161 | 8 |

161 – 17 = 144 | 9 |

144 – 19 = 125 | 10 |

125 – 21 = 104 | 11 |

104 – 23 = 81 | 12 |

81 – 25 = 56 | 13 |

56 – 27 = 29 | 14 |

29 – 29 = 0 | 15 |

**Example 6**: Find the square root of 169 by the method of repeated subtraction

Starting with 169, subtract successive odd numbers until reaching zero. The number of subtractions performed is 13, so the square root of 169 is 13.

Number | Sequence |
---|---|

169 – 1 = 168 | 1 |

168 – 3 = 165 | 2 |

165 – 5 = 160 | 3 |

160 – 7 = 153 | 4 |

153 – 9 = 144 | 5 |

144 – 11 = 133 | 6 |

133 – 13 = 120 | 7 |

120 – 15 = 105 | 8 |

105 – 88 = 88 | 9 |

88 – 19 = 69 | 10 |

69 – 21 = 48 | 11 |

48 – 23 = 25 | 12 |

25 – 25 = 0 | 13 |

### How to Find the Square Root by Division Method

The long division method is commonly used to find square roots of larger numbers. It involves a step-by-step process of dividing the given number into smaller parts and finding the square root digit by digit. This method requires precision and practice to execute accurately.

**Example 1**: Find the square root of 81 using the long division method.

To find the square root of 81 using the long division method, follow these steps:

**Step 1**: Start by grouping the digits of the number into pairs from right to left. For 81, we have 8 and 1.

**Step 2**: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 8. In this case, ‘x’ is 2 because 2 * 2 = 4, which is less than 8.

**Step 3**: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 2.

**Step 4**: Subtract the product of the digit ‘x’ and itself (2 * 2 = 4) from the first pair of digits (8 – 4 = 4). Write the result below the line.

**Step 5**: Bring down the next pair of digits, which is 1.

**Step 6**: Double the current digit on top and write it next to the result. In this case, 2 doubled 4. Write 4 next to the 4 on the line.

**Step 7**: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 41. In this case, ‘y’ is 9 because 49 * 9 = 441, which is less than 410.

**Step 8**: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 29.

**Step 9**: Repeat steps 4 to 8 until all the digits have been brought down and processed.

**Step 10**: Once the division is complete, the result is the square root of the number. In this case, the square root of 81 is 9.

Therefore, the square root of 81 using the long division method is 9.

**Example 2**: Find the square root of 144 using the long division method.

To find the square root of 144 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 144, we have 1 and 44.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 1. In this case, ‘x’ is 1 because 1 * 1 = 1, which is less than 1.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 1.

Step 4: Subtract the product of the digit ‘x’ and itself (1 * 1 = 1) from the first pair of digits (1 – 1 = 0). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 44.

Step 6: Double the current digit on top and write it next to the result. In this case, 1 doubled is 2. Write 2 next to the 0 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 244. In this case, ‘y’ is 4 because 204 * 4 = 816, which is less than 2440.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 14.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 144 is 12.

Therefore, the square root of 144 using the long division method is 12.

**Example 3**: Find the square root of 256 using the long division method.

To find the square root of 256 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 256, we have 25 and 6.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 25. In this case, ‘x’ is 5 because 5 * 5 = 25, which is equal to 25.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 5.

Step 4: Subtract the product of the digit ‘x’ and itself (5 * 5 = 25) from the first pair of digits (25 – 25 = 0). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 6.

Step 6: Double the current digit on top and write it next to the result. In this case, 5 doubled is 10. Write 10 next to the 0 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 60. In this case, ‘y’ is 6 because 106 * 6 = 636, which is less than 600.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 56.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 256 is 16.

Therefore, the square root of 256 using the long division method is 16.

**Example 4**: Find the square root of 625 using the long division method.

To find the square root of 625 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 625, we have 6 and 25.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 6. In this case, ‘x’ is 2 because 2 * 2 = 4, which is less than 6.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 2.

Step 4: Subtract the product of the digit ‘x’ and itself (2 * 2 = 4) from the first pair of digits (6 – 4 = 2). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 25.

Step 6: Double the current digit on top and write it next to the result. In this case, 2 doubled is 4. Write 4 next to the 2 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 225. In this case, ‘y’ is 5 because 245 * 5 = 1225, which is greater than 225.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 25.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 625 is 25.

Therefore, the square root of 625 using the long division method is 25.

**Example 5**: Find the square root of 1024 using the long division method.

To find the square root of 1024 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 1024, we have 10 and 24.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 10. In this case, ‘x’ is 3 because 3 * 3 = 9, which is less than 10.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 3.

Step 4: Subtract the product of the digit ‘x’ and itself (3 * 3 = 9) from the first pair of digits (10 – 9 = 1). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 24.

Step 6: Double the current digit on top and write it next to the result. In this case, 3 doubled is 6. Write 6 next to the 1 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 124. In this case, ‘y’ is 2 because 126 * 2 = 252, which is greater than 124.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 32.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 1024 is 32.

Therefore, the square root of 1024 using the long division method is 32.

**Example 6**: Find the square root of 2401 by the division method

To find the square root of 2401 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 2401, we have 24 and 01.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 24. In this case, ‘x’ is 4 because 4 * 4 = 16, which is less than 24.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 4.

Step 4: Subtract the product of the digit ‘x’ and itself (4 * 4 = 16) from the first pair of digits (24 – 16 = 8). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 01.

Step 6: Double the current digit on top and write it next to the result. In this case, 4 doubled is 8. Write 8 next to the 8 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 801. In this case, ‘y’ is 9 because 898 * 9 = 8082, which is greater than 801.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 49.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 2401 is 49.

Therefore, the square root of 2401 using the long division method is 49.

**Example 7**: Find the square root of 2304 by the division method

To find the square root of 2304 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 2304, we have 23 and 04.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 23. In this case, ‘x’ is 4 because 4 * 4 = 16, which is less than 23.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 4.

Step 4: Subtract the product of the digit ‘x’ and itself (4 * 4 = 16) from the first pair of digits (23 – 16 = 7). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 04.

Step 6: Double the current digit on top and write it next to the result. In this case, 4 doubled is 8. Write 8 next to the 7 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 784. In this case, ‘y’ is 9 because 798 * 9 = 7182, which is greater than 784.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 49.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 2304 is 48.

Therefore, the square root of 2304 using the long division method is 48.

**Example 8**: Find the square root of 4489 by the division method

To find the square root of 4489 using the long division method, follow these steps:

Step 1: Start by grouping the digits of the number into pairs from right to left. For 4489, we have 44 and 89.

Step 2: Find the largest digit ‘x’ such that when ‘x’ is multiplied by itself, the result is less than or equal to 44. In this case, ‘x’ is 6 because 6 * 6 = 36, which is less than 44.

Step 3: Write the digit ‘x’ as the first digit of the square root. So, the square root starts with 6.

Step 4: Subtract the product of the digit ‘x’ and itself (6 * 6 = 36) from the first pair of digits (44 – 36 = 8). Write the result below the line.

Step 5: Bring down the next pair of digits, which is 89.

Step 6: Double the current digit on top and write it next to the result. In this case, 6 doubled is 12. Write 12 next to the 8 on the line.

Step 7: Find the largest digit ‘y’ that can be placed at the end of the current result, such that when the new number formed by the current result and ‘y’ is multiplied by ‘y’, the result is less than or equal to 1289. In this case, ‘y’ is 3 because 12893 * 3 = 38679, which is less than 4489.

Step 8: Write the digit ‘y’ as the next digit of the square root. So, the square root now becomes 63.

Step 9: Repeat steps 4 to 8 until all the digits have been brought down and processed.

Step 10: Once the division is complete, the result is the square root of the number. In this case, the square root of 4489 is 67.

Therefore, the square root of 4489 using the long division method is 67.

### Find Square Root by Estimation Method

Estimation is a quick and convenient approach to finding square roots. It involves approximating the square root based on the closest perfect square. By comparing the given number to known perfect squares, we can make an educated guess and refine our estimate to get closer to the actual square root.

Here are five examples of finding the square root of numbers using the estimation method:

**Example 1**: Find the square root of 17 using the estimation method.

Step 1: Start with an initial guess. Let’s say the initial guess is 4.

Step 2: Square the initial guess: 4 * 4 = 16.

Step 3: Compare the result with the original number. 16 is less than 17.

Step 4: Increase the initial guess by 1: 4 + 1 = 5.

Step 5: Square the new guess: 5 * 5 = 25.

Step 6: Compare the result with the original number. 25 is greater than 17.

Step 7: The square root of 17 lies between 4 and 5. We can estimate it as approximately 4.1.

**Example 2**: Find the square root of 38 using the estimation method.

Step 1: Start with an initial guess. Let’s say the initial guess is 6.

Step 2: Square the initial guess: 6 * 6 = 36.

Step 3: Compare the result with the original number. 36 is less than 38.

Step 4: Increase the initial guess by 1: 6 + 1 = 7.

Step 5: Square the new guess: 7 * 7 = 49.

Step 6: Compare the result with the original number. 49 is greater than 38.

Step 7: The square root of 38 lies between 6 and 7. We can estimate it as approximately 6.2.

**Example 3**: Find the square root of 55 using the estimation method.

Step 1: Start with an initial guess. Let’s say the initial guess is 7.

Step 2: Square the initial guess: 7 * 7 = 49.

Step 3: Compare the result with the original number. 49 is less than 55.

Step 4: Increase the initial guess by 1: 7 + 1 = 8.

Step 5: Square the new guess: 8 * 8 = 64.

Step 6: Compare the result with the original number. 64 is greater than 55.

Step 7: The square root of 55 lies between 7 and 8. We can estimate it as approximately 7.4.

**Example 4**: Find the square root of 72 using the estimation method.

Step 1: Start with an initial guess. Let’s say the initial guess is 8.

Step 2: Square the initial guess: 8 * 8 = 64.

Step 3: Compare the result with the original number. 64 is less than 72.

Step 4: Increase the initial guess by 1: 8 + 1 = 9.

Step 5: Square the new guess: 9 * 9 = 81.

Step 6: Compare the result with the original number. 81 is greater than 72.

Step 7: The square root of 72 lies between 8 and 9. We can estimate it as approximately 8.5.

**Example 5**: Find the square root of 91 using the estimation method.

Step 1: Start with an initial guess. Let’s say the initial guess is 9.

Step 2: Square the initial guess: 9 * 9 = 81.

Step 3: Compare the result with the original number. 81 is less than 91.

Step 4: Increase the initial guess by 1: 9 + 1 = 10.

Step 5: Square the new guess: 10 * 10 = 100.

Step 6: Compare the result with the original number. 100 is greater than 91.

Step 7: The square root of 91 lies between 9 and 10. We can estimate it as approximately 9.5.

Remember, the estimation method provides approximate values of square roots. For more accurate results, other methods such as long division or calculators can be used.