Square Root of 3249

The square root is the inverse of the square of the number. 3249 is a perfect square. The square root of 3249 is expressed in both radical and exponential form. In the radical form, it is expressed as √3249, whereas (3249)^(1/2) in the exponential form. √3249 = 57, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method can be used for perfect square numbers like 3249. However, other methods such as the long-division method and approximation method are also useful. 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 3249 is broken down into its prime factors.

Step 1: Finding the prime factors of 3249 Breaking it down, we get 3 x 3 x 19 x 19: 3² x 19²

Step 2: Now that we have found the prime factors of 3249, the second step is to make pairs of those prime factors. Since 3249 is a perfect square, the digits of the number can be grouped into pairs: (3 x 3) x (19 x 19). Therefore, the square root of 3249 using prime factorization is 3 x 19 = 57.

The long division method is particularly useful for both perfect and non-perfect square numbers. 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 3249, we need to group it as 32 and 49.

Step 2: Now we need to find n whose square is less than or equal to 32. We can say n is ‘5’ because 5 x 5 is 25, which is less than 32. Now the quotient is 5, and after subtracting 25 from 32, the remainder is 7.

Step 3: Now let us bring down 49, which is the new dividend. Add the old divisor with the same number 5 + 5, we get 10, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 10n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 10n × n ≤ 749, let us consider n as 7, now 107 x 7 = 749.

Step 6: Subtract 749 from 749; the difference is 0, and the quotient is 57.

Step 7: Since the remainder is zero, the process stops here. So the square root of √3249 is 57.

The approximation method is another method for finding the square roots, and 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 3249 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √3249. The smallest perfect square less than 3249 is 3249 itself. √3249 is exactly 57.

Step 2: Since 3249 is a perfect square, there is no need for further approximation. The square root of 3249 is exactly 57.

• Square root: A square root is the inverse of a square. Example: 5² = 25, and the inverse of the square is the square root, which is √25 = 5.

• Rational number: A rational number is a number that can 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, but it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 3249 is a perfect square because it is 57 squared.

• Long division method: A systematic way of dividing a number to find the square root, particularly useful for both perfect and non-perfect squares.