Square Root of 3281

The square root is the inverse of the square of a number. 3281 is not a perfect square. The square root of 3281 is expressed in both radical and exponential form. In radical form, it is expressed as √3281, whereas in exponential form it is (3281)^(1/2). The value of √3281 is approximately 57.266, 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 the 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 3281 is broken down into its prime factors.

Step 1: Finding the prime factors of 3281. Breaking it down, we get 7 x 7 x 67: 7^2 x 67.

Step 2: Now that we have found the prime factors of 3281, the second step is to make pairs of those prime factors. Since 3281 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √3281 using prime factorization 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 3281, we need to group it as 32 and 81.

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

Step 3: Now let us bring down 81, which is the new dividend. Add the old divisor with the same number 5 + 5 to 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 ≤ 781. Let us consider n as 7; now 107 x 7 = 749.

Step 6: Subtract 781 from 749; the difference is 32, and the quotient is 57.

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 3200.

Step 8: Now we need to find the new divisor that is 574 because 574 x 4 = 2296.

Step 9: Subtracting 2296 from 3200, we get the result 904.

Step 10: Now the quotient is 57.2.

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 √3281 is approximately 57.27.

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 3281 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3281. The smallest perfect square less than 3281 is 3249, and the largest perfect square greater than 3281 is 3364. √3281 falls somewhere between 57 and 58.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (3281 - 3249) ÷ (3364 - 3249) = 32 ÷ 115 ≈ 0.278. 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 57 + 0.278 = 57.278, so the square root of 3281 is approximately 57.28.

• Square root: A square root is the inverse of a square. Example: 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.

• 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, but the positive square root is often used due to its applications in the real world. This is known as the principal square root.

• Prime factorization: Breaking down a number into its prime number components, which can be multiplied to obtain the original number.

• Long division method: A method used to find the square root of a non-perfect square number by dividing the number into smaller, more manageable parts.