Square Root of 3721

The square root is the inverse of the square of the number. 3721 is a perfect square. The square root of 3721 is expressed in both radical and exponential form. In the radical form, it is expressed as √3721, whereas (3721)^(1/2) in the exponential form. √3721 = 61, 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 is used for perfect square numbers. Since 3721 is a perfect square, the prime factorization method can be applied. Let us now learn the methods:

• Prime factorization method
• Long division method

The product of prime factors is the prime factorization of a number. Now let us look at how 3721 is broken down into its prime factors:

Step 1: Finding the prime factors of 3721. 3721 can be expressed as 61 × 61, as 61 is a prime number. Step 2: Now we found out the prime factors of 3721. Since 3721 is a perfect square, the digits of the number can be grouped in pairs.

Therefore, the square root of 3721 using prime factorization is 61.

The long division method is particularly used for both perfect and non-perfect square numbers to ensure accuracy. 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 pairs. For 3721, we group it as 37 and 21.

Step 2: Now we need to find a number whose square is less than or equal to 37. We can choose 6 because 6 × 6 = 36. Now the quotient is 6, after subtracting 36 from 37, the remainder is 1.

Step 3: Bring down the next pair of digits (21) to make the new dividend 121.

Step 4: Double the quotient obtained in the previous step (6) and place it as a new divisor part: 12.

Step 5: Find a digit (n) such that 12n × n is less than or equal to 121. Choose n as 1 because 121 × 1 = 121.

Step 6: Subtract 121 from 121 to get a remainder of 0.

The quotient is 61, which is the square root of 3721.

Students may make mistakes while finding the square root, such as not recognizing the number as a perfect square or miscalculating in the long division steps. Here are a few common mistakes and how to avoid them:

• Square root: A square root is the inverse of a square. Example: 8² = 64, and the inverse of the square is the square root that is √64 = 8.
   
• 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 36 is a perfect square of 6.
   
• Prime factorization: The process of expressing a number as the product of its prime numbers.
   
• Long division method: A method to find the square root of a number, particularly useful for both perfect and non-perfect squares through a step-by-step process.