Square Root of 12345

The square root is the inverse of the square of the number. 12345 is not a perfect square. The square root of 12345 is expressed in both radical and exponential form. In the radical form, it is expressed as √12345, whereas (12345)^(1/2) in the exponential form. √12345 ≈ 111.10806, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the Prime factorization of a number. Now let us look at how 12345 is broken down into its prime factors.

Step 1: Finding the prime factors of 12345 Breaking it down, we get 3 x 5 x 823.

Step 2: Now we found out the prime factors of 12345. The second step is to make pairs of those prime factors. Since 12345 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 12345 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 12345, we need to group it as 45 and 123.

Step 2: Now, we need to find n whose square is closest to 123. We can say n as '11' because 11 x 11 = 121, which is closest to 123. Now the quotient is 11, and after subtracting 123 - 121, the remainder is 2.

Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number, 11 + 11, to get 22, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and the quotient. Now we get 22n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 22n × n ≤ 245. Let us consider n as 1, now 22 x 1 x 1 = 22.

Step 6: Subtract 245 from 22, the difference is 223, and the quotient is 111.

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 zeros to the dividend. Now the new dividend is 22300.

Step 8: Now we need to find the new divisor that is 222 because 2221 x 1 = 2221.

Step 9: Subtracting 2221 from 22300, we get the result 2079.

Step 10: Now the quotient is 111.1.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √12345 ≈ 111.10

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 12345 using the approximation method.

Step 1: Now we have to find the closest perfect square of √12345. The smallest perfect square less than 12345 is 12100, and the largest perfect square greater than 12345 is 14400. √12345 falls somewhere between 110 and 120.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) ÷ (Largest perfect square - smallest perfect square)

Going by the formula (12345 - 12100) ÷ (14400 - 12100) = 0.10625

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 110 + 0.10625 ≈ 110.11, so the square root of 12345 is approximately 111.11.

• 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.

• Long division method: A method used to find the square root of non-perfect square numbers through division and approximation.

• Prime factorization: The process of breaking down a number into its basic prime factors.

• 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.