Square Root of 638

The square root is the inverse operation of squaring a number. 638 is not a perfect square. The square root of 638 is expressed in both radical and exponential form. In the radical form, it is expressed as √638, whereas in exponential form it is (638)^(1/2). √638 ≈ 25.25866, 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 typically used for perfect square numbers. However, for non-perfect squares like 638, methods such as the long-division method and approximation method are preferred. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as a product of its prime factors. Let's explore this for 638:

Step 1: Find the prime factors of 638. Breaking it down, we have 2 x 11 x 29. The prime factorization is 2^1 x 11^1 x 29^1.

Step 2: Since 638 is not a perfect square, its prime factors cannot be paired evenly.

Therefore, finding the square root of 638 via prime factorization is not straightforward.

The long division method is suitable for non-perfect square numbers. Here's how to find the square root of 638 using this method:

Step 1: Group the digits of 638 from right to left. Here, we have 38 and 6.

Step 2: Find the largest number n whose square is less than or equal to 6. This is 2, as 2 x 2 = 4 ≤ 6. The quotient is 2, and the remainder is 6 - 4 = 2.

Step 3: Bring down the next pair of digits (38) to get 238. Double the divisor (2) to get 4, and form the new divisor 4n.

Step 4: Find n such that 4n x n ≤ 238. If n = 5, 45 x 5 = 225.

Step 5: Subtract 225 from 238 to get 13. The quotient becomes 25.

Step 6: Add a decimal, and bring down two zeros to make the dividend 1300.

Step 7: Double the quotient (25) to get 50. Find n such that 50n x n ≤ 1300. If n = 2, 502 x 2 = 1004.

Step 8: Subtract 1004 from 1300 to get 296. Step 9: Continue this process to get more decimal places.

The approximate square root of 638 is 25.258.

The approximation method is another way to find square roots. Here's how to approximate the square root of 638:

Step 1: Identify the perfect squares closest to 638. These are 625 (25^2) and 676 (26^2). Thus, √638 is between 25 and 26.

Step 2: Use the formula for approximation: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (638 - 625) / (676 - 625) = 13 / 51 ≈ 0.2549. Adding this to 25 gives 25.2549.

Thus, √638 ≈ 25.258.

• Square root: The square root is the inverse operation of squaring a number. For example, if 4^2 = 16, then √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction of two integers. For example, √638 is an irrational number.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Prime factorization: Expressing a number as a product of its prime factors. For example, the prime factorization of 638 is 2 x 11 x 29.
   
• Long division method: A method to find square roots of non-perfect squares by using a division-like process to find digits of the square root one at a time.