Square Root of 1853

The square root is the inverse of the square of the number. 1853 is not a perfect square. The square root of 1853 is expressed in both radical and exponential form. In radical form, it is expressed as √1853, whereas (1853)^(1/2) in the exponential form. √1853 ≈ 43.057, 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:

• Prime factorization method

• Long division method

• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 1853 is broken down into its prime factors.

Step 1: Finding the prime factors of 1853 Breaking it down, we get 1853 = 3 x 617 (both 3 and 617 are prime numbers).

Step 2: Since 1853 is not a perfect square, calculating it using prime factorization alone is not feasible, as the digits cannot be grouped in pairs.

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 1853, we need to group it as 53 and 18.

Step 2: Now we need to find n whose square is less than or equal to 18. We can say n as ‘4’ because 4 x 4 = 16 is less than or equal to 18. Now the quotient is 4 and after subtracting 16 from 18, the remainder is 2.

Step 3: Now let us bring down 53 which is the new dividend. Add the old divisor with the same number 4 + 4 = 8 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 253. Let us consider n as 3, now 83 x 3 = 249.

Step 6: Subtract 253 from 249, the difference is 4, and the quotient is 43.

Step 7: Since the dividend is less than the divisor, we can add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 400.

Step 8: Now we need to find the new divisor that is 860 because 860 x 0 = 0 and 0 < 400.

Step 9: Subtracting 0 from 400, the result is 400.

Step 10: Now the quotient is 43.0

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

So the square root of √1853 is approximately 43.057.

The approximation method is another method for finding square roots, and 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 1853 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1853. The smallest perfect square less than 1853 is 1764 (42^2) and the largest perfect square greater than 1853 is 1936 (44^2). √1853 falls somewhere between 42 and 44.

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 (1853 - 1764) / (1936 - 1764) = 89 / 172 ≈ 0.517.

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 42 + 0.517 = 42.517, so the square root of 1853 is approximately 43.057.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16 and the inverse of squaring 16 is the square root, which 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 expression of a number as the product of its prime factors is called prime factorization. For example, the prime factorization of 30 is 2 x 3 x 5.

• 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.