Square Root of 7680

The square root is the inverse of the square of the number. 7680 is not a perfect square. The square root of 7680 is expressed in both radical and exponential form. In the radical form, it is expressed as √7680, whereas (7680)^(1/2) in exponential form. √7680 ≈ 87.5885, 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 7680 is broken down into its prime factors.

Step 1: Finding the prime factors of 7680 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 8: 2^7 x 3 x 5^2

Step 2: Now we found the prime factors of 7680. The second step is to make pairs of those prime factors. Since 7680 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 7680 using prime factorization is not straightforward.

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 7680, we need to group it as 80 and 76.

Step 2: Now we need to find n whose square is close to 76. We can say n as ‘8’ because 8 x 8 = 64, which is lesser than 76. Now the quotient is 8, after subtracting 76 - 64, the remainder is 12.

Step 3: Now let us bring down 80, which is the new dividend. Add the old divisor with the same number 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 16n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 16n x n ≤ 1280. Let us consider n as 7, now 16 x 7 x 7 = 784.

Step 6: Subtract 1280 from 784, the difference is 496, and the quotient is 87.

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 49600.

Step 8: Now we need to find the new divisor that is 175 because 1755 x 5 = 8775.

Step 9: Subtracting 8775 from 49600, we get the result 40825.

Step 10: Now the quotient is 87.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values continue till the remainder is zero. So, the square root of √7680 is approximately 87.59.

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 7680 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7680. The smallest perfect square less than 7680 is 7569, and the largest perfect square greater than 7680 is 7744. √7680 falls somewhere between 87 and 88.

Step 2: Now we need to apply the linear approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (7680 - 7569) / (7744 - 7569) = 0.61. 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 87 + 0.61 = 87.61. So, the square root of 7680 is approximately 87.61.

• 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 the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: The expression of a number as a product of its prime factors.
   
• Approximation: A method used to find a value that is close enough to the correct value, typically within some margin of error.