Square Root of 657

The square root is the inverse operation of squaring a number. Since 657 is not a perfect square, its square root is an irrational number. The square root of 657 is expressed in both radical and exponential forms. In radical form, it is expressed as √657, whereas in exponential form it is (657)^(1/2). The approximate value of √657 is 25.629, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers, but not for non-perfect square numbers like 657. Instead, the long-division method or approximation method is used. Let us now explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number involves breaking it down into the product of its prime factors. However, since 657 is not a perfect square, it cannot be simplified through prime factorization alone. Let's see its breakdown:

Step 1: Finding the prime factors of 657 Breaking it down, we get 3 x 3 x 73: 3^2 x 73

Step 2: Since 657 is not a perfect square, its prime factors cannot be perfectly paired.

Thus, calculating √657 using only prime factorization is not possible.

The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root of 657 using the long division method, step by step:

Step 1: Group the digits from right to left. For 657, we group it as 57 and 6.

Step 2: Find n such that n^2 is the largest perfect square less than or equal to 6. Here, n is 2 because 2^2 = 4.

Step 3: Subtract 4 from 6 to get a remainder of 2, and bring down 57, making the new dividend 257.

Step 4: Double the divisor (which is 2), and use it as the first part of the new divisor, getting 4n. Find n such that 4n × n is less than or equal to 257. Here, n is 5, since 45 × 5 = 225.

Step 5: Subtract 225 from 257 to get a remainder of 32. Extend the division by adding decimals and zeros as necessary.

Step 6: Continue repeating these steps until the desired precision is achieved.

Thus, the square root of 657 is approximately 25.629.

The approximation method is another way to estimate square roots. Here's how to approximate the square root of 657:

Step 1: Identify the closest perfect squares around 657. The closest perfect squares are 625 (25^2) and 676 (26^2).

Step 2: Since 657 is between these two perfect squares, √657 falls between 25 and 26.

Step 3: Use the formula: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square) Applying the formula: (657 - 625) / (676 - 625) = 32 / 51 ≈ 0.627 Add this decimal to 25, giving an approximation of 25.627 for √657.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse is √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction, where the denominator is not zero, and both the numerator and denominator are integers.
   
• Principal square root: The principal square root is the non-negative square root of a number, used in most real-world applications.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, 657 = 3 × 3 × 73.
   
• Long division method: A technique used to find the square root of a number by dividing the number into parts and estimating the square root iteratively.