Square Root of 821

The square root is the inverse of the square of the number. 821 is not a perfect square. The square root of 821 is expressed in both radical and exponential form.

In the radical form, it is expressed as √821, whereas (821)^(1/2) in the exponential form. √821 ≈ 28.63624, 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 821 is broken down into its prime factors.

Step 1: Finding the prime factors of 821 Breaking it down, we get 821 = 3 x 7 x 39.

Step 2: Now we found out the prime factors of 821. The second step is to make pairs of those prime factors. Since 821 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 821 using prime factorization is impossible.

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 821, we need to group it as 21 and 8.

Step 2: Now we need to find n whose square is 8. We can say n as ‘2’ because 2 x 2 is less than or equal to 8. Now the quotient is 2; after subtracting 8 - 4, the remainder is 4.

Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number 2 + 2 to 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, we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 421. Let us consider n as 6, now 46 x 6 = 276.

Step 6: Subtract 421 from 276, the difference is 145, and the quotient is 26.

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 14500.

Step 8: Now we need to find the new divisor that is 57 because 573 x 3 = 1719.

Step 9: Subtracting 1719 from 14500, we get the result 12781.

Step 10: Now the quotient is 28.6.

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 √821 is approximately 28.63.

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 821 using the approximation method.

Step 1: Now we have to find the closest perfect square of √821. The smallest perfect square less than 821 is 784, and the largest perfect square more than 821 is 841. √821 falls somewhere between 28 and 29.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (821 - 784) / (841 - 784) ≈ 0.65.

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 28 + 0.65 = 28.65, so the square root of 821 is approximately 28.65.

• Square root: A square root is the inverse of a square. For 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 the reason it is also known as the principal square root.
   
• Prime factorization: This is the process of expressing a number as the product of its prime factors.
   
• Long division method: A method used for finding the square root of numbers, especially useful for non-perfect squares, by breaking down the number into more manageable parts.