Square Root of 1286

The square root is the inverse of the square of a number. 1286 is not a perfect square. The square root of 1286 is expressed in both radical and exponential form. In the radical form, it is expressed as √1286, whereas in exponential form it is (1286)^(1/2). √1286 ≈ 35.851, 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 1286 is broken down into its prime factors.

Step 1: Finding the prime factors of 1286.Breaking it down, we get 2 x 643.

Step 2: Now we have found the prime factors of 1286. Since 1286 is not a perfect square, we cannot pair the digits of the number. Therefore, calculating the square root of 1286 using prime factorization is not feasible.

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: Group the numbers from right to left. In the case of 1286, we need to group it as 28 and 12.

Step 2: Find n whose square is less than or equal to 12. We can say n is 3 because 3^2 = 9, which is less than 12. Subtract 9 from 12, and the remainder is 3.

Step 3: Bring down 86, which makes the new dividend 386. Add the old divisor with the same number 3 + 3 to get 6 as the new divisor.

Step 4: Find 6n × n ≤ 386. Consider n as 6, now 66 x 6 = 396, which is more than 386, so try with n as 5.

Step 5: 65 x 5 = 325, which is less than 386. Subtract 325 from 386, and the remainder is 61.

Step 6: Since the dividend is less than the divisor, add a decimal point, allowing us to add two zeros to the dividend. The new dividend is 6100.

Step 7: Find the new divisor, which is 715, because 715 x 9 = 6435.

Step 8: Subtract 6435 from 6100 to get the result -335.

Step 9: Continue these steps until we get two numbers after the decimal point. If there is no remainder, continue until the remainder is zero.

So the square root of √1286 ≈ 35.85.

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 1286 using the approximation method.

Step 1: Find the closest perfect square of √1286. The smallest perfect square less than 1286 is 1225, and the largest perfect square greater than 1286 is 1369. √1286 falls between 35 and 37.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1286 - 1225) / (1369 - 1225) ≈ 0.626. Adding this to the initial estimate, we get 35 + 0.85 ≈ 35.85, so the square root of 1286 is approximately 35.85.

• Square root: A square root is the inverse of a square. Example: 4² = 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.

• Principal square root: A number has both positive and negative square roots. However, the positive square root is more prominent due to its practical uses, hence known as the principal square root.

• Prime factorization: Breaking down a composite number into a product of its prime numbers. For example, the prime factorization of 1286 is 2 × 643.

• Long division method: A step-by-step method used to find the square root of a non-perfect square number by dividing, multiplying, and subtracting in a sequential manner.