Square Root of 12500

The square root is the inverse of the square of the number. 12500 is not a perfect square. The square root of 12500 is expressed in both radical and exponential form. In the radical form, it is expressed as √12500, whereas (12500)^(1/2) in the exponential form. √12500 ≈ 111.8034, 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 12500 is broken down into its prime factors:

Step 1: Finding the prime factors of 12500 Breaking it down, we get 2 x 2 x 5 x 5 x 5 x 5 x 5: 2² x 5⁵

Step 2: Now we found the prime factors of 12500. The second step is to make pairs of those prime factors. Since 12500 is not a perfect square, some factors will remain unpaired.

The prime factorization can simplify √12500 = √(2² x 5⁵) = 5² x √(2 x 5) = 25 x √10 ≈ 111.8034.

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 12500, we need to group it as 12 | 500.

Step 2: Now we need to find a number n whose square is less than or equal to 12. We can say n is '3' because 3 x 3 = 9, which is less than 12. Now the quotient is 3, and after subtracting 12 - 9, the remainder is 3.

Step 3: Bring down 50, making the new dividend 350. Double the quotient 3 to get 6, which will be part of our new divisor.

Step 4: Find a digit x such that the number 6x multiplied by x is less than or equal to 350. In this case, 65 x 5 = 325.

Step 5: Subtract 325 from 350, getting a remainder of 25. Bring down the next pair of zeroes, making it 2500.

Step 6: The new divisor is 65 + 5 = 70. Finding a digit x for 70x x x ≤ 2500, we find 703 x 3 = 2109.

Step 7: Subtract 2109 from 2500, getting a remainder of 391. Continue the process with more decimal places as needed. So, the square root of 12500 using the long division method is approximately 111.80.

The approximation method is another method for finding square roots, and 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 12500 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √12500. The smallest perfect square less than 12500 is 12100 (110²) and the largest perfect square greater than 12500 is 14400 (120²). √12500 falls somewhere between 110 and 120.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula: (12500 - 12100) / (14400 - 12100) = 400 / 2300 ≈ 0.1739 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 + 0.1739 ≈ 111.80, so the square root of 12500 is approximately 111.80.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, which 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.

• Prime factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 12500 is 2² x 5⁵.

• Long division method: A method used for finding the square root of non-perfect squares by dividing and finding a quotient iteratively.

• Approximation method: A method of finding the square root by estimating between the nearest perfect squares and using interpolation.