Square Root of 626

The square root is the inverse of the square of the number. 626 is not a perfect square. The square root of 626 is expressed in both radical and exponential form. In the radical form, it is expressed as √626, whereas (626)^(1/2) in the exponential form. √626 ≈ 25.01999, 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 long-division method and approximation methods 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. Now let us look at how 626 is broken down into its prime factors.

Step 1: Finding the prime factors of 626 Breaking it down, we get 2 x 313: 2^1 x 313^1

Step 2: Now we have found the prime factors of 626. Since 626 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 626 using prime factorization is not feasible.

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 626, we need to group it as 26 and 6.

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

Step 3: Now let us bring down 26, which is the new dividend. 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, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 226. Let us consider n as 5; now 4 x 5 x 5 = 225.

Step 6: Subtract 225 from 226; the difference is 1, 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 100.

Step 8: Now we need to find a new divisor that is 250, because 250 x 0 = 0.

Step 9: Subtracting 0 from 100, we get the result 100. Step 10: The next digit in the quotient is 0.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √626 ≈ 25.02

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 626 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √626. The smallest perfect square less than 626 is 625, and the largest perfect square greater than 626 is 676. √626 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). Going by the formula: (626 - 625) / (676 - 625) = 1 / 51 ≈ 0.0196 Using the formula, we identified the decimal part of our square root. The next step is adding the value we got initially to the decimal number, which is 25 + 0.0196 ≈ 25.02. So the square root of 626 is approximately 25.02.

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

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.

• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.