Square Root of 628

The square root is the inverse of the square of a number. 628 is not a perfect square. The square root of 628 is expressed in both radical and exponential form. In the radical form, it is expressed as √628, whereas 628^(1/2) is the exponential form. √628 ≈ 25.0599, 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. However, for non-perfect square numbers like 628, the long division method and approximation method are used. Let us now learn these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 628 is broken down into its prime factors.

Step 1: Finding the prime factors of 628. Breaking it down, we get 2 x 2 x 157: 2^2 x 157^1.

Step 2: Now we found out the prime factors of 628. The second step is to make pairs of those prime factors. Since 628 is not a perfect square, the digits of the number cannot be grouped into pairs.

Therefore, calculating √628 using prime factorization alone is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 628, we need to group it as 28 and 6.

Step 2: Now we need to find n whose square is less than or equal to 6. We can say n as ‘2’ because 2 × 2 = 4, which is less than 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Bring down 28, making the new dividend 228. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 228. Let us consider n as 5; now 45 x 5 = 225.

Step 6: Subtract 225 from 228; the difference is 3, and the quotient is 25.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 300.

Step 8: Now we need to find the new divisor that is 501 because 501 × 0.5 = 250.5.

Step 9: Subtracting 250.5 from 300, we get the result 49.5.

Step 10: Now the quotient is 25.0.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So the square root of √628 is approximately 25.06.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 628 using the approximation method.

Step 1: Now we have to find the closest perfect square of √628. The smallest perfect square less than 628 is 625 (25^2), and the largest perfect square greater than 628 is 676 (26^2). √628 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Applying the formula: (628 - 625) ÷ (676 - 625) = 3 ÷ 51 ≈ 0.059. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 25 + 0.059 ≈ 25.059.

So the square root of 628 is approximately 25.059.

• Square root: A square root is the inverse of a square. For example, 5^2 = 25 and the inverse of the square is the square root, so √25 = 5.
   
• Irrational number: An irrational number is a number that cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots, but the positive square root is more commonly used and is known as the principal square root.
   
• Prime factorization: The process of breaking down a composite number into its prime factors. For example, the prime factorization of 28 is 2 x 2 x 7.
   
• Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups of two digits starting from the right.