Square Root of 1226

The square root is the inverse of the square of the number. 1226 is not a perfect square. The square root of 1226 is expressed in both radical and exponential form. In the radical form, it is expressed as √1226, whereas (1226)^(1/2) in the exponential form. √1226 ≈ 35.01428, 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 1226 is broken down into its prime factors.

Step 1: Finding the prime factors of 1226 Breaking it down, we get 2 × 613

Step 2: Now we found out the prime factors of 1226. Since 1226 is not a perfect square, calculating √1226 using prime factorization directly is not feasible.

The long division method is particularly used for non-perfect square numbers. 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 1226, we need to group it as 26 and 12.

Step 2: Find n whose square is less than or equal to 12. We can say n as 3 because 3 × 3 = 9 is less than 12. Now the quotient is 3 after subtracting 9 from 12, the remainder is 3.

Step 3: Bring down 26, making it the new dividend, 326. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be 6n (where n is part of the quotient). We find n such that 6n × n ≤ 326. Step 5: Let n = 5, then 65 × 5 = 325.

Step 6: Subtract 325 from 326, the difference is 1, and the quotient is 35.

Step 7: Since the dividend is smaller 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 100.

Step 8: Calculate the new divisor, 700, by adding a decimal number to the quotient and repeating the steps to get more decimal places.

So the square root of √1226 is approximately 35.01.

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 1226 using the approximation method.

Step 1: Find the closest perfect squares to √1226. The smallest perfect square less than 1226 is 1225, and the largest perfect square greater than 1226 is 1296. √1226 falls between 35 and 36.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (1226 - 1225) / (1296 - 1225) ≈ 0.014 Adding this decimal to the smaller perfect square root gives 35 + 0.014 = 35.014, so the square root of 1226 is approximately 35.014.

• 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, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed 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 commonly used, known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Long division method: A method used to find the square root of non-perfect squares through a systematic division process.