Square Root of 28.13

The square root is the inverse of the square of a number. 28.13 is not a perfect square. The square root of 28.13 is expressed in both radical and exponential form. In the radical form, it is expressed as √28.13, whereas (28.13)^(1/2) is its exponential form. √28.13 ≈ 5.303, 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, the prime factorization method is not applicable for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. However, since 28.13 is not an integer, it cannot be expressed in terms of prime factors. Therefore, the prime factorization method is not applicable in this case.

The long division method is particularly used for non-perfect square numbers. In this method, we can estimate the square root by performing a series of divisions:

Step 1: Start by grouping the digits from right to left. For 28.13, treat it as 28.1300 for ease of calculation.

Step 2: Find the largest number whose square is less than or equal to 28. The number is 5, since 5 × 5 = 25. The remainder is 28 - 25 = 3.

Step 3: Bring down the next pair of digits (13) to make the dividend 313.

Step 4: Double the quotient (5) and find a new digit (n) such that 10n × n ≤ 313. The suitable digit is 3, since 103 × 3 = 309. The remainder is 313 - 309 = 4.

Step 5: Bring down the next pair of digits (00) to make the dividend 400.

Step 6: Double the current quotient (53) to get 106 and find n such that 106n × n ≤ 400, which is 3, since 1063 × 3 = 318. The remainder is 400 - 318 = 82.

Step 7: The quotient now reads 5.303.

The approximation method is another method for finding square roots, providing a quick way to estimate the square root of a given number.

Step 1: Identify the closest perfect squares around 28.13. The closest perfect squares are 25 (5²) and 36 (6²).

Step 2: Since 28.13 is closer to 25 than to 36, the square root of 28.13 will be slightly more than 5.

Step 3: Use interpolation to approximate: (28.13 - 25) / (36 - 25) ≈ (3.13 / 11) ≈ 0.284

Step 4: Add 0.284 to 5, resulting in approximately 5.284.

• Square root: A square root is the inverse of a square. For example, 5² = 25, and the inverse of the square is the square root, √25 = 5.
   
• Irrational number: An irrational number cannot be written as a simple fraction (p/q) where p and q are integers and q ≠ 0.
   
• Principal square root: The principal square root refers to the positive square root of a number.
   
• Long division method: A systematic method to find the square root of non-perfect squares, involving a series of division steps.
   
• Approximation method: A method used to estimate the square root of non-perfect squares by identifying nearby perfect squares and interpolating.