Square Root of 1323

The square root is the inverse of the square of the number. 1323 is not a perfect square. The square root of 1323 is expressed in both radical and exponential form. In the radical form, it is expressed as √1323, whereas (1323)^(1/2) in the exponential form. √1323 ≈ 36.371, 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 1323 is broken down into its prime factors.

Step 1: Finding the prime factors of 1323

Breaking it down, we get 3 x 3 x 3 x 3 x 3 x 3 x 11: 3^3 x 3^3 x 11

Step 2: Now we found out the prime factors of 1323. The second step is to make pairs of those prime factors. Since 1323 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √1323 using prime factorization is not straightforward.

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 1323, we need to group it as 23 and 13.

Step 2: Now we need to find n whose square is 13. We can say n as ‘3’ because 3 x 3 is less than or equal to 13. The quotient is 3 after subtracting 9 from 13, 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 3 + 3, we get 6 which will be our new divisor.

Step 4: The new divisor will be the sum of the previous quotient and divisor. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 423. Let us consider n as 7, now 67 x 7 = 469

Step 6: Subtract 469 from 423, and the remainder is -46, indicating we need to adjust n and repeat the process.

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 4600.

Step 8: Now we need to find the new divisor that is 72 because 72 x 72 = 5184, which is too large.

Step 9: Adjusting again, we find 71 x 71 = 5041, which is still too large.

Step 10: We continue this process until we get a closer approximation.

So the square root of √1323 is approximately 36.371.

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 1323 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1323. The smallest perfect square less than 1323 is 1296 and the largest perfect square greater than 1323 is 1369. √1323 falls somewhere between 36 and 37.

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 (1323 - 1296) ÷ (1369 - 1296) = 27 ÷ 73 ≈ 0.36986. 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 36 + 0.37 = 36.37, so the square root of 1323 is approximately 36.37.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 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: It is the expression of a number as the product of its prime factors. For example, the prime factorization of 1323 is 3^3 x 11.

• Long division method: A step-by-step process of division used to find the square root of non-perfect square numbers, involving grouping numbers and finding approximate values.