Square Root of 1282

The square root is the inverse of the square of a number. 1282 is not a perfect square. The square root of 1282 is expressed in both radical and exponential forms. In radical form, it is expressed as √1282, whereas in exponential form it is (1282)^(1/2). √1282 ≈ 35.7931, 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 1282 is broken down into its prime factors.

Step 1: Finding the prime factors of 1282 Breaking it down, we get 2 × 641.

Step 2: Now we found out the prime factors of 1282. The second step is to make pairs of those prime factors. Since 1282 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1282 using prime factorization is not feasible for finding its square root.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers for the given number. 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 1282, we need to group it as 28 and 12.

Step 2: Now we need to find n whose square is 12. We can say n is ‘3’ because 3 × 3 is less than or equal to 12. Now the quotient is 3; after subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 28, making the new dividend 328. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 328. Let us consider n as 5, now 65 × 5 = 325.

Step 6: Subtracting 325 from 328, the difference is 3, and the quotient is 35.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 300.

Step 8: Now we need to find the new divisor, which is 705, because 705 × 5 = 3525.

Step 9: Subtracting 3525 from 3000, we get the result -525.

Step 10: Now the quotient is 35.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √1282 is approximately 35.79.

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 1282 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1282. The smallest perfect square less than 1282 is 1225, and the largest perfect square greater than 1282 is 1369. √1282 falls somewhere between 35 and 37.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1282 - 1225) / (1369 - 1225) = 57 / 144 = 0.3958. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 35 + 0.3958 = 35.3958, so the square root of 1282 is approximately 35.39.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which is √16 = 4.
   
• Irrational number: An irrational number 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, it is usually the positive square root that is more widely used in practical applications.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Long division method: A step-by-step process used to find the square root of non-perfect squares by dividing the number into smaller, more manageable parts.