Square Root of 14600

The square root is the inverse of the square of the number. 14600 is not a perfect square. The square root of 14600 is expressed in both radical and exponential form. In the radical form, it is expressed as √14600, whereas (14600)^(1/2) in the exponential form. √14600 ≈ 120.83, 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 typically used for non-perfect square numbers where long-division and approximation methods are utilized. 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 14600 is broken down into its prime factors.

Step 1: Finding the prime factors of 14600 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 73: 2³ x 5² x 73

Step 2: Now we found the prime factors of 14600. The second step is to make pairs of those prime factors. Since 14600 is not a perfect square, calculating its square root using prime factorization directly is not feasible.

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 14600, we need to group it as 00, 46, and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n as ‘1’ because 1 x 1 is equal to 1. Now the quotient is 1, and after subtracting 1 - 1, the remainder is 0.

Step 3: Now let us bring down 46, which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 46. Let us consider n as 2, now 2 x 2 x 2 = 44.

Step 6: Subtract 46 from 44; the difference is 2.

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 zeroes to the dividend. Now the new dividend is 200.

Step 8: Now we need to find the new divisor. Continuing with this process will eventually lead to an approximate value of the square root, which is 120.83.

The approximation method is another way of 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 14600 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √14600. The smallest perfect square less than 14600 is 14400 (120²), and the largest perfect square greater than 14600 is 14884 (122²). √14600 falls somewhere between 120 and 122.

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

Using the formula (14600 - 14400) ÷ (14884 - 14400) = 0.83

Adding the decimal to the initial integer estimate gives us 120 + 0.83 = 120.83, so the square root of 14600 is approximately 120.83.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse square root of 16 is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers. Its decimal form is non-terminating and non-repeating, such as √14600.

• Perfect square: A perfect square is the product of an integer multiplied by itself. For example, 144 is a perfect square of 12.

• Approximation: This refers to finding an estimated value or result that is close to the actual one, often used when dealing with irrational numbers.

• Long division: A method used to find the square root of numbers that are not perfect squares, involving division and subtraction processes.