Square Root of 823

The square root is the inverse of the square of the number. 823 is not a perfect square. The square root of 823 is expressed in both radical and exponential form.

In the radical form, it is expressed as √823, whereas (823)^(1/2) in the exponential form. √823 ≈ 28.68074, 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 823 is broken down into its prime factors.

Step 1: Finding the prime factors of 823 Since 823 is a prime number, it cannot be broken down into smaller prime factors other than itself.

Step 2: As 823 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 823 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 823, we need to group it as 23 and 8.

Step 2: Now we need to find n whose square is close to 8. We can say n is '2' because 2 × 2 is lesser than or equal to 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Now let us bring down 23, 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: Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 423. Let us consider n as 6, now 46 × 6 = 276.

Step 6: Subtract 276 from 423; the difference is 147, 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 14700.

Step 8: Now we need to find the new divisor which is 36 because 536 × 6 = 3216.

Step 9: Subtracting 3216 from 14700, we get the result 11484.

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 √823 is approximately 28.68.

The approximation method is another method for finding the square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 823 using the approximation method.

Step 1: Now we have to find the closest perfect square of √823. The smallest perfect square less than 823 is 784 and the largest perfect square greater than 823 is 841. √823 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 (823 - 784) ÷ (841 - 784) ≈ 0.6807.

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 + 0.6807 ≈ 28.6807, so the square root of 823 is approximately 28.6807.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, so √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 has no positive divisors other than 1 and itself.
   
• 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.
   
• Approximation: Approximation is the process of finding a value that is close enough to the correct value, typically within an acceptable range of error.