Square Root of 7920

The square root is the inverse of the square of the number. 7920 is not a perfect square. The square root of 7920 is expressed in both radical and exponential form. In the radical form, it is expressed as √7920, whereas (7920)^(1/2) in the exponential form. √7920 ≈ 88.9833, 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 7920 is broken down into its prime factors.

Step 1: Finding the prime factors of 7920 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 5 x 11: 2^4 x 3^2 x 5 x 11

Step 2: Now we have found the prime factors of 7920. The second step is to make pairs of those prime factors. Since 7920 is not a perfect square, complete pairing is impossible. Therefore, calculating √7920 using prime factorization alone 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 7920, we need to group it as 20 and 79.

Step 2: Now we need to find n whose square is less than or equal to 79. We can say n as ‘8’ because 8 x 8 = 64 is less than or equal to 79. Now the quotient is 8, subtract 64 from 79, and the remainder is 15.

Step 3: Now let us bring down 20, making the new dividend 1520. Add the old divisor with the same number 8 + 8, giving us 16 as the new divisor

Step 4: The new divisor will be the number formed by appending n to 16, making it 168. Now we need to find the value of n such that 168n x n is less than or equal to 1520.

Step 5: The next step is finding 168n x n ≤ 1520. Let us consider n as 9, now 168 x 9 = 1512, which is less than 1520.

Step 6: Subtract 1512 from 1520, the difference is 8, and the quotient is 89.

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 800.

Step 8: Continue this process until you achieve the desired level of accuracy. So the square root of √7920 is approximately 88.98.

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 7920 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7920. The smallest perfect square below 7920 is 7840 (88^2) and the largest perfect square above 7920 is 8100 (90^2). √7920 falls somewhere between 88 and 90.

Step 2: Now we need to apply the approximation formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (7920 - 7744) ÷ (8100 - 7744) = 176 ÷ 356 ≈ 0.4944 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 88 + 0.4944 ≈ 88.49.

• 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.
   
• Approximation method: This method is used to estimate the square roots of non-perfect squares by identifying nearby perfect squares.
   
• Long division method: A method used to find the square root of non-perfect squares through iterative division and subtraction.
   
• Prime factorization: The process of decomposing a number into its constituent prime numbers.