Square Root of 14900

The square root is the inverse of the square of the number. 14900 is not a perfect square. The square root of 14900 is expressed in both radical and exponential form. In the radical form, it is expressed as √14900, whereas (14900)^(1/2) in the exponential form. √14900 ≈ 122.0655, 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 14900 is broken down into its prime factors.

Step 1: Finding the prime factors of 14900 Breaking it down, we get 2 x 2 x 5 x 5 x 149: 2² x ⁵² x 149

Step 2: Now we have found the prime factors of 14900. The second step is to make pairs of those prime factors. Since 14900 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √14900 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 14900, we need to group it as (1)(49)(00).

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

Step 3: Now let us bring down 49, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be 20n. We need to find the largest digit n such that 20n x n is less than or equal to 49. Let us consider n as 2, now 20 x 2 x 2 = 40.

Step 5: Subtract 49 from 40; the difference is 9. We bring down the next group, which is 00, making the new dividend 900.

Step 6: Now, double the quotient obtained so far to get 24, and find the new tentative divisor 240 + n such that (240 + n) x n ≤ 900. We find n as 3 because 243 x 3 = 729.

Step 7: Subtract 900 from 729 to get a remainder of 171.

Step 8: 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 17100.

Step 9: Now we need to find the new divisor, which is 244n + n, because 2446 x 6 = 14676.

Step 10: Subtracting 14676 from 17100, we get the result 2424.

Step 11: Continue doing these steps until we get the desired level of precision after the decimal point.

So the square root of √14900 is approximately 122.0655.

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 14900 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √14900. The smallest perfect square less than 14900 is 14400 (120²), and the largest perfect square more than 14900 is 15625 (125²). √14900 falls between 120 and 125.

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

Applying this formula: (14900 - 14400) / (15625 - 14400) = 500 / 1225 ≈ 0.408 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 120 + 0.408 ≈ 120.408. After further refining and checking, we find it approximates to 122.0655.

• 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, 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: Prime factorization is the process of expressing a number as the product of its prime factors.

• Long Division Method: A step-by-step process used to divide a number into smaller parts to find the square root.