Square Root of 1285

The square root is the inverse of the square of the number. 1285 is not a perfect square. The square root of 1285 is expressed in both radical and exponential form. In the radical form, it is expressed as √1285, whereas in exponential form, it is expressed as (1285)^(1/2). √1285 ≈ 35.83165, 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 typically used for perfect square numbers. However, for non-perfect square numbers, the long division and approximation methods are preferred. Let's explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors constitutes the prime factorization of a number. Let us look at how 1285 is broken down into its prime factors:

Step 1: Finding the prime factors of 1285 Breaking it down, we get 5 x 257. Since 1285 is not a perfect square, the prime factors cannot be paired evenly.

Thus, calculating the square root of 1285 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. Let's 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 1285, group it as 12 and 85.

Step 2: Find the largest number whose square is less than or equal to 12. This number is 3, as 3 x 3 = 9. Subtract 9 from 12, resulting in a remainder of 3.

Step 3: Bring down the next pair of digits, 85, to make the new dividend 385. Double the divisor (3), making it 6.

Step 4: Find a number 'n' such that 6n x n ≤ 385. The appropriate number is 5, since 65 x 5 = 325.

Step 5: Subtract 325 from 385, resulting in 60.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeroes, making the new dividend 6000.

Step 7: The new divisor is 705. Find 'n' such that 705n x n ≤ 6000. The appropriate number is 8, since 705 x 8 = 5640.

Step 8: Subtracting 5640 from 6000 results in 360. Step 9: Continue this process until a satisfactory level of precision is achieved.

The square root of 1285 is approximately 35.83.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Let's learn how to find the square root of 1285 using the approximation method:

Step 1: Identify the perfect squares closest to 1285. The smallest perfect square is 1225 (35^2), and the largest is 1369 (37^2). Therefore, √1285 falls between 35 and 37.

Step 2: Use the formula: (Given number - smallest perfect square) / (largest perfect square - smallest perfect square). For 1285: (1285 - 1225) / (144) = 60 / 144 = 0.4167. Add this decimal to the smaller square root: 35 + 0.4167 = 35.4167. So the approximate square root of 1285 is 35.83.

• Square root: The square root is the inverse of squaring a number. For example, if 4^2 = 16, then the square root is √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction p/q, where p and q are integers and q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots, but the positive root is often used in real-world applications and is known as the principal square root.
   
• Approximation: An estimated value close to the actual value, used when precise calculation is complex.
   
• Long division method: A systematic method of finding the square root of a number by breaking it down into steps, similar to long division.