Square Root of 1155

The square root is the inverse operation of squaring a number. 1155 is not a perfect square. The square root of 1155 can be expressed in both radical and exponential forms. In radical form, it is expressed as √1155, whereas in exponential form it is (1155)^(1/2). The square root of 1155 is approximately 33.985, which is an irrational number because it cannot be expressed as a fraction of two integers.

For non-perfect square numbers like 1155, methods such as the long division method and approximation method are used to find the square root. Let's explore these methods:

• Prime factorization method 
   
• Long division method
   
• Approximation method

Prime factorization involves expressing a number as a product of its prime factors. Now, let us see how 1155 is broken down into its prime factors:

Step 1: Finding the prime factors of 1155. Breaking it down, we get 3 x 5 x 7 x 11 = 3^1 x 5^1 x 7^1 x 11^1.

Step 2: Since 1155 is not a perfect square, pairs of prime factors cannot be formed. Therefore, finding the square root of 1155 using prime factorization is not feasible for simplification.

The long division method is used for non-perfect square numbers. This method involves finding the closest perfect square numbers and using them to approximate the square root. Let's see how to apply this method:

Step 1: Start by grouping the numbers from right to left. For 1155, group as 11 and 55.

Step 2: Find the largest number whose square is less than or equal to 11. Here, it is 3, since 3 × 3 = 9. Subtract 9 from 11, leaving a remainder of 2.

Step 3: Bring down the next group, 55, to make it 255. Double the divisor (3) to get 6, and find the next digit in the quotient by seeing what fits, i.e., 66n ≤ 255.

Step 4: We find n as 3 since 66 × 3 = 198. Subtract 198 from 255 to get 57.

Step 5: Continue the process of bringing down zeros and repeating steps until the desired precision is achieved. The square root of 1155 is approximately 33.985.

The approximation method is another straightforward way to estimate the square root of a number. Let's see how to find the square root of 1155 using this method: Step 1: Identify the nearest perfect squares around 1155. The closest are 1089 (33^2) and 1156 (34^2). Thus, √1155 is between 33 and 34. Step 2: Use the formula for approximation: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (1155 - 1089) / (1156 - 1089) = 66/67 ≈ 0.985. Step 3: Add this decimal to the smaller perfect square root: 33 + 0.985 = 33.985. Thus, √1155 ≈ 33.985.

Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, since 5 × 5 = 25. Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, such as √1155, which has a non-repeating, non-terminating decimal expansion. Prime factorization: This involves expressing a number as a product of its prime numbers. For example, the prime factorization of 1155 is 3 × 5 × 7 × 11. Long division method: A method used to find the square root of non-perfect square numbers through a series of division steps. Approximation: A method of estimating the value of a square root by using nearby perfect squares or other means to achieve a reasonable estimate.