Square Root of 1129

The square root is the inverse of the square of the number. 1129 is not a perfect square. The square root of 1129 is expressed in both radical and exponential form. In the radical form, it is expressed as √1129, whereas (1129)^(1/2) in the exponential form. √1129 ≈ 33.603, 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 1129 is broken down into its prime factors.

Step 1: Finding the prime factors of 1129 Breaking it down, we get 1129 as a prime number itself, as it is not divisible by any prime number up to its square root.

Step 2: Since 1129 is not a perfect square, calculating its square root using prime factorization directly is not feasible.

The long division method is particularly used for non-perfect square numbers. 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 1129, we can group it as 11 and 29.

Step 2: Now we need to find n whose square is closest to 11. In this case, n is 3 because 3 × 3 = 9, which is less than 11. The quotient is 3, and after subtracting 9 from 11, the remainder is 2.

Step 3: Bring down 29, making the new dividend 229. Double the quotient (3) to get 6, which will be part of our new divisor.

Step 4: Find a digit x such that 6x × x is less than or equal to 229. The digit is 3, so 63 × 3 = 189.

Step 5: Subtract 189 from 229, leaving a remainder of 40. Add a decimal point and pair of zeros to the dividend, making it 4000.

Step 6: Double the digits of the current quotient to get 66 and find a digit y such that 66y × y is less than or equal to 4000. The digit is 6.

Step 7: Subtract the product from 4000, and the process continues until the desired precision is reached. The final value approximates √1129 ≈ 33.603.

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 1129 using the approximation method.

Step 1: Identify the closest perfect squares around 1129. The smallest perfect square is 1024 (32^2), and the largest is 1156 (34^2). √1129 falls between 32 and 34.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (1129 - 1024) / (1156 - 1024) = 105 / 132 ≈ 0.795.

Step 3: Add this decimal to the lower boundary of the square root range: 32 + 0.795 ≈ 32.795. Therefore, the square root of 1129 is approximately 33.603.

• Square root: A square root is the inverse of squaring a number. Example: If 5^2 = 25, then √25 = 5.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction. Examples include √2 and √1129.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Example: 1129.

• Decimal: A decimal is a number that includes a whole number and a fractional part separated by a decimal point. Examples include 3.14 and 33.603.

• Long division method: A method used to find the square root of non-perfect squares by grouping digits and finding each digit of the square root iteratively.