Square Root of 3649

The square root is the inverse of the square of the number. 3649 is not a perfect square. The square root of 3649 is expressed in both radical and exponential form. In the radical form, it is expressed as √3649, whereas (3649)^(1/2) in the exponential form. √3649 ≈ 60.374, 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 used for non-perfect square numbers where 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. Now let us look at how 3649 is broken down into its prime factors.

Step 1: Finding the prime factors of 3649 3649 is a prime number, so its only prime factors are 1 and itself.

Step 2: Since 3649 is not a perfect square and cannot be expressed with repeated prime factors, calculating √3649 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 3649, we need to group it as 49 and 36.

Step 2: Now we need to find n whose square is 36. We can say n as ‘6’ because 6 × 6 = 36. Now the quotient is 6 and the remainder is 0.

Step 3: Now let us bring down 49 which is the new dividend. Add the old divisor with the same number, 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 49. Let us consider n as 4, now 12 × 4 = 48.

Step 6: Subtract 49 from 48, the difference is 1, and the quotient is 60.

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 the new divisor, which is 120, because 120 × 0 = 0.

Step 9: Subtracting 0 from 100 we get the result 100.

Step 10: Now the quotient is 60.3.

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

So the square root of √3649 is approximately 60.37.

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 3649 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3649. The smallest perfect square less than 3649 is 3600 (60^2), and the largest perfect square greater than 3649 is 3721 (61^2). √3649 falls somewhere between 60 and 61.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (3649 - 3600) ÷ (3721 - 3600) = 49 ÷ 121 ≈ 0.39. 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 60 + 0.39 ≈ 60.39.

So the square root of 3649 is approximately 60.39.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Example: 2, 3, 5, 7, 11, etc.
   
• 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.
   
• Long division method: A method used to find the square root of a number by dividing it into pairs of digits, starting from the right, and finding the largest number whose square is less than or equal to the number being divided.