## Methods to Calculate the Square Root of 24

There are two primary methods to find the square root of 24:

### Method 1: Prime Factorisation

Using the prime factorisation method, we can express:

24 = 2 × 2 × 2 × 3

This can be rewritten as:

24 = (2 x 2) x 2 x 3

24 = 2^{2} x 2 x 3 (By laws of exponents)

Taking square root on both sides, we get:

\(\begin{array}{l}\sqrt{24} = \sqrt{2^{2}\times 2\times 3}\end{array} \)

Taking the square term out of the root, we get:

√24 = 2√6

This is the simplest form of the square root of 24.

We can further put the value of √6 and simplify.

Since, the approximate value of √6 = 2.5

Therefore, **√24 = 2√6 = 2 x 2.5 ≈ 4.9**

**Key Facts:**

**The square root of 24 is an irrational number.****24 is an even composite number.****24 is not a perfect square, so √24 is not a natural number.****The roots of 24 are +2√6 or -2√6.**

### Method 2: Long Division

The long division method is another accurate way to find the square root of any number. It is a shortcut method that can be easily learnt and applied by following these steps:

- Step 1: Write the number 24 as 24.00 00 00 00
- Step 2: Group the digits in pairs of two and place a bar above each pair
- Step 3: Choose a number that, when multiplied by itself, results in a number less than or equal to 24. For example, 4 x 4 = 16
- Step 4: Subtract 16 from 24 to get 8. Add 4 to the previous divisor to get 8. Bring down the two zeros and write them next to 8 on the dividend side.
- Step 5: Again, choose a number and place it in the unit place of the divisor such that the new divisor multiplied by that number is less than or equal to 800. For example, 88 x 8 = 704
- Step 6: Repeat the above steps to obtain the quotient up to two decimal places.
- The final result is 4.89, as shown in the diagram below:

You can repeat these steps to find the square root value up to four decimal places.

### Approximation Method

The approximation method can also be used to find the square root of imperfect squares. It provides a close approximate value to the actual square root. This method is particularly useful for one or two-digit numbers. Given below are the square roots of perfect numbers from 1 to 10:

1 | 1 |

2 | 4 |

3 | 9 |

4 | 16 |

5 | 25 |

6 | 36 |

7 | 49 |

8 | 64 |

9 | 81 |

10 | 100 |

Now, it's clear that the number 24 lies between the squared terms 16 and 25, or between 4^{2} and 5^{2}. Therefore, 24 is the square of a number less than 5^{2}. Hence, the first digit of √24 is 4.

To find the digits after the decimal:

Decimal part = (Actual number - Lower perfect square) / (Higher perfect square - Lower perfect square)

= (24-16)/(25-16)

= 8/9

= 0.88888…

Therefore, the value of √24 = 4.888888…

Or √24 = 4.89

Hence, we have the correct answer.

### Repeated Subtraction Method

The repeated subtraction method is another way of finding the square root. However, it only applies to perfect squares. Since 24 is not a perfect square, we cannot use this method to find its square root.

In the repeated subtraction method, we subtract consecutive odd numbers starting from the original number until we get zero. The total number of times the subtraction is done gives us the square root value. If we attempt to use this method to find the square root of 24, we get:

- 24 – 1 = 23
- 23 – 3 = 20
- 20 – 5 = 15
- 15 – 7 = 8
- 8 – 9 = -1

Since we obtained a negative integer instead of zero, we cannot find the square root of 24 using this method.

### Square Roots of Various Numbers

√20 | 4.472 |

√21 | 4.583 |

√22 | 4.690 |

√23 | 4.796 |

√24 | 4.899 |

√25 | 5.000 |

√26 | 5.099 |

√27 | 5.196 |

√28 | 5.292 |

√29 | 5.385 |