Square Root of 1889

The square root is the inverse of squaring a number. 1889 is not a perfect square. The square root of 1889 is expressed in both radical and exponential form. In radical form, it is expressed as √1889, whereas in exponential form, it is (1889)^(1/2). √1889 ≈ 43.4483, which is an irrational number because it cannot be expressed as p/q, where p and q are integers and q ≠ 0.

The prime factorization method is typically used for perfect squares. However, for non-perfect squares, methods like 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 prime factorization of a number involves expressing it as a product of prime factors. Now let us see how 1889 is broken down into its prime factors:

Step 1: Finding the prime factors of 1889. It turns out 1889 is a prime number itself, so it cannot be broken down further into prime factors.

Step 2: Since 1889 is a prime number and not a perfect square, calculating its square root using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. Here is how to find the square root using the long division method, step by step:

Step 1: Group the numbers from right to left. For 1889, group it as 18 and 89.

Step 2: Find n whose square is less than or equal to 18. We choose n as 4 because 4 x 4 = 16. The quotient is 4, and the remainder is 2 after subtracting 16 from 18.

Step 3: Bring down 89 to make the new dividend 289. Double the quotient for the new divisor, 4 + 4 = 8.

Step 4: Find a digit x such that 8x multiplied by x is less than or equal to 289. The correct digit is 3, as 83 x 3 = 249.

Step 5: Subtract 249 from 289 to get a remainder of 40.

Step 6: Add a decimal point to the quotient and bring down 00 to make the new dividend 4000.

Step 7: The new divisor is 86x. Find x such that 86x multiplied by x is less than or equal to 4000. The digit is 4, giving 864 x 4 = 3456.

Step 8: Subtract 3456 from 4000 to get a remainder of 544.

Step 9: The quotient so far is 43.4. Continue the process to obtain more decimal places if needed.

So the square root of √1889 is approximately 43.448.

The approximation method is another way to find square roots. It's an easy method to find the square root of a given number. Now let's learn how to find the square root of 1889 using this method.

Step 1: Find the closest perfect squares around 1889. The smallest perfect square less than 1889 is 1764 (42^2), and the largest perfect square greater than 1889 is 1936 (44^2). √1889 falls between 42 and 44.

Step 2: Apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

By the formula, (1889 - 1764) / (1936 - 1764) = 125 / 172 ≈ 0.7267.

The initial approximation is 42 + 0.7267 ≈ 42.7267. Refine further to get √1889 ≈ 43.4483.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16 and the inverse of the square is the square root, so √16 = 4.

• Irrational number: An irrational number cannot be expressed as a fraction p/q, where q ≠ 0 and p and q are integers.

• Prime number: A number greater than 1 that has no positive divisors other than 1 and itself.

• Radical: The symbol √ used to denote the square root or nth root of a number.

• Long division method: A technique for finding the square roots of non-perfect squares through a step-by-step division process.