Square Root of 14450

The square root is the inverse of the square of the number. 14450 is not a perfect square. The square root of 14450 is expressed in both radical and exponential forms. In the radical form, it is expressed as √14450, whereas (14450)^(1/2) in the exponential form. √14450 ≈ 120.184, 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 and approximation methods 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 14450 is broken down into its prime factors.

Step 1: Finding the prime factors of 14450 Breaking it down, we get 2 x 5² x 17 x 17: 2 x 5² x 17 x 17.

Step 2: Now we found out the prime factors of 14450. The second step is to make pairs of those prime factors. Since 14450 is not a perfect square, the digits of the number can’t be grouped into a complete pair.

Therefore, calculating √14450 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 14450, we need to group it as 50 and 144.

Step 2: Now we need to find n whose square is close to or slightly less than 144. We can say n is ‘12’ because 12 x 12 = 144, which is equal to 144. Now the quotient is 12, and the remainder is 0.

Step 3: Now let us bring down 50, which is the new dividend. Add the old divisor with the same number; 12 + 12 = 24, which will be our new divisor.

Step 4: The next step is finding 24n × n ≤ 50. Let us consider n as 2, now 24 x 2 = 48.

Step 5: Subtract 50 from 48, and the difference is 2. The quotient now is 12.2.

Step 6: 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. The new dividend is 200.

Step 7: Now we need to find the new divisor that is 244 because 244 x 1 = 244, which is more than 200. Adjust accordingly to get as close as possible.

Step 8: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero. So the square root of √14450 ≈ 120.184.

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 14450 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √14450. The smallest perfect square below 14450 is 14400 (120²), and the largest perfect square above 14450 is 14641 (121²). √14450 falls somewhere between 120 and 121.

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

Applying the formula, (14450 - 14400) ÷ (14641 - 14400) ≈ 0.184. 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: 120 + 0.184 ≈ 120.184, so the square root of 14450 is approximately 120.184.

• Square root: A square root is the inverse of a square. For 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 the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3².

• 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.