Square Root of 1744

The square root is the inverse of the square of the number. 1744 is not a perfect square. The square root of 1744 is expressed in both radical and exponential form. In the radical form, it is expressed as √1744, whereas (1744)^(1/2) in the exponential form. √1744 ≈ 41.758, 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 1744 is broken down into its prime factors.

Step 1: Finding the prime factors of 1744 Breaking it down, we get 2 x 2 x 2 x 2 x 109: 2^4 x 109^1

Step 2: Now we have found the prime factors of 1744. The second step is to make pairs of those prime factors. Since 1744 is not a perfect square, the digits of the number can’t be grouped in pairs to form a perfect square.

Therefore, calculating √1744 using prime factorization requires approximation.

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 1744, we need to group it as 44 and 17.

Step 2: Now we need to find n whose square is ≤ 17. We can say n is ‘4’ because 4 x 4 = 16 is less than or equal to 17. Now the quotient is 4, and after subtracting 16 from 17, the remainder is 1.

Step 3: Now let us bring down 44, 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 8n, and we need to find the value of n such that 8n x n ≤ 144. Let us consider n as 1, now 81 x 1 = 81.

Step 5: Subtract 81 from 144, the difference is 63, and the quotient is 41.

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. Now the new dividend is 6300.

Step 7: Now we need to find the new divisor, which is 418 because 418 x 8 = 3344.

Step 8: Subtracting 3344 from 6300, we get the result 2956.

Step 9: Continue doing these steps until we get the desired number of decimal places.

So the square root of √1744 is approximately 41.758.

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 1744 using the approximation method.

Step 1: Now we have to find the closest perfect square to √1744.

The smallest perfect square less than 1744 is 1600, and the largest perfect square greater than 1744 is 1764.

√1744 falls somewhere between 40 and 42.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula: (1744 - 1600) / (1764 - 1600) = 144 / 164 ≈ 0.878.

Using the formula, we identified the decimal point of our square root.

The next step is adding the integer part, which is 40, to the decimal number we found: 40 + 0.878 = 40.878.

Thus, the approximate square root of 1744 is 40.878.

• 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.
   
• Prime factorization: Breaking down a number into its basic prime number factors.
   
• Long division method: A step-by-step process of dividing a number to find its square root, especially useful for non-perfect squares.
   
• Approximation method: A method used to estimate the square root of a number by comparing it to the nearest perfect squares.