Square Root of 1280

The square root is the inverse of the square of the number. 1280 is not a perfect square. The square root of 1280 is expressed in both radical and exponential form. In the radical form, it is expressed as √1280, whereas (1280)^(1/2) in the exponential form. √1280 ≈ 35.7771, 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 prime factorization method is not used; instead, 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 1280 is broken down into its prime factors.

Step 1: Finding the prime factors of 1280. Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5: 2^7 x 5^2.

Step 2: Now we found out the prime factors of 1280. The second step is to make pairs of those prime factors. Since 1280 is not a perfect square, therefore the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating √1280 using prime factorization involves using the pairs and the unpaired factor.

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: To begin with, we need to group the numbers from right to left. In the case of 1280, we need to group it as 80 and 12.

Step 2: Now we need to find n whose square is 12. We can say n as ‘3’ because 3 x 3 is lesser than or equal to 12. Now the quotient is 3 after subtracting 9 (3^2) from 12, the remainder is 3.

Step 3: Now let us bring down 80, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: Find a number n such that 6n x n ≤ 380. Let us consider n as 5, now 65 x 5 = 325.

Step 5: Subtract 325 from 380; the difference is 55, and the quotient is 35.

Step 6: 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 5500.

Step 7: Now we need to find the new divisor that is 707 because 707 x 7 = 4949.

Step 8: Subtracting 4949 from 5500, we get the result 551.

Step 9: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √1280 ≈ 35.77.

The approximation method is another method for finding the 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 1280 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1280. The smallest perfect square of 1280 is 1225, and the largest perfect square of 1280 is 1296. √1280 falls somewhere between 35 and 36.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1280 - 1225) ÷ (1296 - 1225) ≈ 0.7771. 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 + 0.7771 ≈ 35.7771.

So the square root of 1280 is approximately 35.7771.

• Square root: A square root is the inverse of a square. Example: 4^2 =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, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Long division method: A systematic method used to divide numbers and find square roots of non-perfect squares through repeated subtraction and division steps.