Square Root of 12850

The square root is the inverse of the square of a number. 12850 is not a perfect square. The square root of 12850 is expressed in both radical and exponential form. In the radical form, it is expressed as √12850, whereas in exponential form it is expressed as (12850)^(1/2). √12850 ≈ 113.362, 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 12850 is broken down into its prime factors:

Step 1: Finding the prime factors of 12850 Breaking it down, we get 2 x 5 x 5 x 257: 2¹ x 5² x 257¹

Step 2: Now we have found the prime factors of 12850. The second step is to make pairs of those prime factors. Since 12850 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √12850 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 12850, we need to group it as 50 and 128.

Step 2: Now we need to find n whose square is ≤ 128. We can say n as ‘11’ because 11 x 11 = 121, which is lesser than 128. Now the quotient is 11, after subtracting 128 - 121, the remainder is 7.

Step 3: Now let us bring down 50 which is the new dividend. Add the old divisor with the same number 11 + 11 = 22, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and 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 ≤ 750. Let us consider n as 3, now 223 x 3 = 669. Step 6: Subtract 750 from 669, the difference is 81, and the quotient is 113.

Step 7: Since the remainder is not zero, 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 8100.

Step 8: Now we need to find the new divisor that is 226 because 226 x 3 = 678.

Step 9: Subtracting 678 from 8100 we get the result 1422.

Step 10: The quotient is 113.3

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

So the square root of √12850 is approximately 113.36.

The approximation method is another method for finding the 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 12850 using the approximation method.

Step 1: Now we have to find the closest perfect square of √12850. The smallest perfect square less than 12850 is 12769 (which is 113²) and the largest perfect square more than 12850 is 12896 (which is 114²). √12850 falls somewhere between 113 and 114.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square) Using the formula (12850 - 12769) / (12896 - 12769) = 81 / 127 = 0.6378

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 113 + 0.64 = 113.64, so the square root of 12850 is approximately 113.64.

• 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: The expression of a number as a product of its prime factors. Example: 18 = 2 x 3².

• Long division method: A technique used to find the square root of a non-perfect square by dividing, multiplying, subtracting, and bringing down numbers in steps.