Square Root of 1729

The square root is the inverse of the square of a number. 1729 is not a perfect square. The square root of 1729 is expressed in both radical and exponential form. In radical form, it is expressed as √1729, whereas in exponential form it is (1729)^(1/2). √1729 ≈ 41.5633, 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, for non-perfect square numbers, 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 1729 is broken down into its prime factors:

Step 1: Finding the prime factors of 1729 Breaking it down, we get 7 x 13 x 19.

Step 2: Now we found out the prime factors of 1729. The second step is to make pairs of those prime factors. Since 1729 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating √1729 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square number to the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Group the numbers from right to left. In the case of 1729, group it as 17 and 29.

Step 2: Find a number whose square is less than or equal to 17. This number is 4, since 4 x 4 = 16. The quotient is 4, and the remainder is 1 (17 - 16).

Step 3: Bring down the next pair, 29, to get the new dividend, 129.

Step 4: Double the previous quotient (4) to get 8, which will be part of the new divisor.

Step 5: Find a number n such that 8n x n is less than or equal to 129. In this case, n is 1, since 81 x 1 = 81.

Step 6: Subtract 81 from 129 to get a remainder of 48, and the quotient is 41.

Step 7: Add decimal points to the quotient and bring down pairs of zeros.

Step 8: Repeat the process to find the next digit(s) after the decimal point.

The square root of 1729 is approximately 41.563.

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 1729 using the approximation method.

Step 1: Find the closest perfect squares around 1729.

The smallest is 1600 (40^2) and the largest is 1764 (42^2).

√1729 falls somewhere between 40 and 42.

Step 2: Apply the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

(1729 - 1600) / (1764 - 1600) = 129 / 164 ≈ 0.786.

Step 3: Add this decimal to the smaller square root: 40 + 0.786 = 40.786.

The square root of 1729 is approximately 41.563, refining the approximation from the previous steps.

• Square root: A square root is the inverse operation to squaring a number. Example: 4^2 = 16, and the inverse of squaring is the square root, so √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction (p/q), where p and q are integers and q is not zero.
   
• Approximation: The process of finding a value that is close enough to the right answer, typically with some thought or calculation involved.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, and 16 are perfect squares.
   
• Long division method: A method used to find the square root of a number by performing division and averaging processes iteratively.