Square Root of 1848

The square root is the inverse of the square of the number. 1848 is not a perfect square. The square root of 1848 is expressed in both radical and exponential form. In the radical form, it is expressed as √1848, whereas (1848)^(1/2) in the exponential form. √1848 ≈ 42.978, 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:

• 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 1848 is broken down into its prime factors.

Step 1: Finding the prime factors of 1848 Breaking it down, we get 2 x 2 x 2 x 3 x 7 x 11: 2^3 x 3 x 7 x 11

Step 2: Now we found out the prime factors of 1848.

The second step is to make pairs of those prime factors.

Since 1848 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1848 using prime factorization is impossible.

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 1848, we need to group it as 48 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. Subtracting 16 from 18, the remainder is 2.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8, which will be part of our new divisor.

Step 4: The new divisor will be 8, and we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 248. Let us consider n as 3, now 83 x 3 = 249.

Step 6: Subtract 249 from 248, the difference is 1, and the quotient is 43.

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 100.

Step 8: Now we need to find the new divisor that is 861 because 861 x 1 = 861.

Step 9: Subtracting 861 from 1000, we get the result 139.

Step 10: Now the quotient is 43.9.

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

So the square root of √1848 is approximately 42.98.

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 1848 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1848. The smallest perfect square less than 1848 is 1764, and the largest perfect square greater than 1848 is 1936. √1848 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).

Going by the formula (1848 - 1764) / (1936 - 1764) = 0.5.

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 + 0.5 = 42.5, so the square root of 1848 is approximately 42.5.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 6^2 = 36, and the square root of 36 is √36 = 6.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction, i.e., it cannot be expressed as p/q, where p and q are integers and q ≠ 0.

• Principal square root: The principal square root of a number is the non-negative square root. For example, the principal square root of 9 is 3.

• Perfect square: A perfect square is an integer that is the square of an integer. Example: 16 is a perfect square because it is 4^2.

• Decimal: A decimal number is a number that contains a decimal point, separating the whole number from the fractional part. Example: 7.86, 8.65, and 9.42 are decimals.