Square Root of 1223

The square root is the inverse of the square of the number. 1223 is not a perfect square. The square root of 1223 is expressed in both radical and exponential form. In the radical form, it is expressed as √1223, whereas (1223)^(1/2) in the exponential form. √1223 ≈ 34.9637, 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 1223 is broken down into its prime factors.

Step 1: Finding the prime factors of 1223 Breaking it down, we get 11 x 111: 11 x 3 x 37

Step 2: Now we found out the prime factors of 1223. The second step is to make pairs of those prime factors. Since 1223 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1223 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 1223, we need to group it as 23 and 12.

Step 2: Now we need to find n whose square is closest to 12. We can say n as ‘3’ because 3 x 3 is lesser than or equal to 12. Now the quotient is 3, and after subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 23, which is the new dividend. Add the old divisor with the same number, 3 + 3, and we get 6, which will be our new divisor.

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

Step 5: The next step is finding 6n × n ≤ 323; let us consider n as 5, now 65 x 5 = 325

Step 6: Since 325 is greater than 323, let us consider n as 4. Now 64 x 4 = 256

Step 7: Subtract 256 from 323; the difference is 67, and the quotient is 34.

Step 8: 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 6700.

Step 9: Now we need to find the new divisor that is 9 because 679 x 9 = 6111

Step 10: Subtracting 6111 from 6700, we get the result 589.

Step 11: Now the quotient is 34.9 Step 12: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √1223 is approximately 34.96.

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 1223 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1223. The smallest perfect square less than 1223 is 1156, and the largest perfect square greater than 1223 is 1296. √1223 falls somewhere between 34 and 36.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1223 - 1156) ÷ (1296 - 1156) = 67 ÷ 140 ≈ 0.4786 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 34 + 0.4786 = 34.4786, so the square root of 1223 is approximately 34.48.

• Square root: A square root is the inverse of a square. 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 a principal square root.
   
• Prime factorization: Breaking down a number into its smallest prime factors.
   
• Approximation: A method of finding an estimate of a number or value when an exact answer is not possible or practical.