Square Root of 13725

The square root is the inverse of the square of the number. 13725 is not a perfect square. The square root of 13725 is expressed in both radical and exponential form. In the radical form, it is expressed as √13725, whereas (13725)^(1/2) in the exponential form. √13725 ≈ 117.1573, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 13725, methods such as the long division method and approximation method are used. Let us now learn these methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 13725 is broken down into its prime factors:

Step 1: Finding the prime factors of 13725 Breaking it down, we get 3 × 3 × 5 × 5 × 61: 3² × 5² × 61

Step 2: Now we found out the prime factors of 13725. The next step is to make pairs of those prime factors. Since 13725 is not a perfect square, some digits of the number can’t be grouped in pairs.

Therefore, calculating the exact square root of 13725 using prime factorization is not possible, but it gives us a simplified radical form.

The long division method is particularly used for non-perfect square numbers. In this method, we find the square root step-by-step. Let's learn how to find the square root using the long division method:

Step 1: Group the numbers from right to left. In the case of 13725, group it as 725 and 13.

Step 2: Find a number whose square is less than or equal to 13. Here, n is 3 because 3 × 3 = 9 ≤ 13. Subtracting 9 from 13 gives a remainder of 4.

Step 3: Bring down the next pair 25. The new dividend is 425.

Step 4: Double the divisor (3) to get 6, which will be part of our new divisor.

Step 5: Find a digit n such that 6n × n ≤ 425. Here, n is 7 because 67 × 7 = 469 > 425, but 66 × 6 = 396 < 425.

Step 6: Subtract 396 from 425 to get 29.

Step 7: Bring down pairs of zeros and repeat the process to find more decimal places. So, the square root of √13725 is approximately 117.1573.

The approximation method helps find the square roots easily by identifying the closest perfect squares. Let's find the square root of 13725 using this method:

Step 1: Identify the closest perfect squares to √13725. The smallest perfect square less than 13725 is 13689 (117²) and the largest perfect square greater than 13725 is 13824 (118²). √13725 falls between 117 and 118.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (13725 - 13689) / (13824 - 13689) ≈ 0.36

Add this decimal to the smaller square root: 117 + 0.36 = 117.36. So, the square root of 13725 is approximately 117.36.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16, and the inverse is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a fraction of two integers. Example: √2, π.

• Principal square root: The non-negative square root of a number is known as the principal square root.

• Prime factorization: Expressing a number as a product of its prime factors. Example: The prime factorization of 18 is 2 × 3².

• Long division method: A method used to find the square root of a number by dividing it into smaller, manageable parts. It is particularly useful for non-perfect squares.