Square Root of 12800

The square root is the inverse of the square of the number. 12800 is not a perfect square. The square root of 12800 is expressed in both radical and exponential form. In the radical form, it is expressed as √12800, whereas (12800)^(1/2) in the exponential form. √12800 = 113.137, 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:

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 12800 is broken down into its prime factors.

Step 1: Finding the prime factors of 12800 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5: 2⁷ x 5⁴

Step 2: Now we found out the prime factors of 12800. The second step is to make pairs of those prime factors. The prime factors can be grouped into pairs: (2⁶) x (5⁴) x 2, resulting in the square root of 12800 being 2³ x 5² x √2.

Therefore, calculating √12800 using prime factorization gives us an approximate value of 113.137.

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 12800, we need to group it as 00 and 128.

Step 2: Now we need to find a number whose square is less than or equal to 128. We can say it as ‘11’ because 11 x 11 is 121, which is less than 128. Now the quotient is 11, and after subtracting 128 - 121, the remainder is 7.

Step 3: Bring down the next pair of digits, 00, making the new dividend 700. Add the old divisor with the same number 11 + 11 to get 22, which will be our new divisor.

Step 4: Find the largest digit n such that 22n x n ≤ 700. Let's consider n as 3, so 223 x 3 = 669.

Step 5: Subtract 669 from 700, and the difference is 31.

Step 6: Since the remainder 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, resulting in 3100.

Step 7: Now find the new divisor, considering 226, whose product with a certain digit results in a number less than or equal to 3100.

Step 8: Continue this process until the desired level of precision is reached. So the square root of √12800 is approximately 113.137.

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 12800 using the approximation method.

Step 1: Now we have to find the closest perfect square to √12800. The smallest perfect square less than 12800 is 12100 (110²), and the largest perfect square is 14400 (120²). √12800 falls somewhere between 110 and 120.

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

Using the formula (12800 - 12100) / (14400 - 12100) ≈ 0.304

Using the formula, we add 0.304 to the smallest integer root, which is 110 + 0.304 = 110.304, which is a rough estimate. The actual square root is closer to 113.137, as calculated more accurately.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 x 4 = 16.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction - its decimal goes on forever without repeating.

• Radical: A radical is a symbol that represents the root of a number. For example, √ is the radical symbol for square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 121 is a perfect square because it is 11 x 11.

• Long division method: A method used to find the square root of numbers that are not perfect squares by dividing the number into pairs and estimating the quotient step by step.