Square Root of 9680

The square root is the inverse of the square of a number. 9680 is not a perfect square. The square root of 9680 is expressed in both radical and exponential form. In the radical form, it is expressed as √9680, whereas in the exponential form, it is expressed as (9680)^(1/2). The square root of 9680 is approximately 98.3906, 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 9680 is broken down into its prime factors.

Step 1: Finding the prime factors of 9680

Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 3 × 5 × 11 × 11: \(2^5 \times 3^1 \times 5^1 \times 11^2\)

Step 2: Now we found out the prime factors of 9680. The second step is to make pairs of those prime factors. Since 9680 is not a perfect square, the digits of the number can’t be grouped into pairs perfectly. Therefore, calculating the exact square root of 9680 using prime factorization is not possible.

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 9680, we need to group it as 80 and 96.

Step 2: Now we need to find a number whose square is less than or equal to 96. We can say the number is 9 because 9 × 9 = 81, which is less than 96. Now, after subtracting 81 from 96, the remainder is 15, and the quotient is 9.

Step 3: Let us bring down 80, making the new dividend 1580. Add the old divisor with the same number, 9 + 9, to get 18, which will be our new divisor.

Step 4: The new divisor is 18n. We need to find the value of n such that 18n × n is less than or equal to 1580. Let us consider n as 8. Now, 188 × 8 = 1504.

Step 5: Subtract 1504 from 1580. The difference is 76, and the quotient is 98.

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 zeros to the dividend. Now the new dividend is 7600.

Step 7: Now we need to find a new divisor. Suppose the new divisor is 1969 because 1969 × 9 = 17721.

Step 8: Subtracting 17721 from 17600 results in a remainder of 879.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose there is no decimal value; continue until the remainder is zero.

So the square root of √9680 is approximately 98.39.

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 9680 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √9680. The closest perfect square below 9680 is 9600 (which is 98^2), and the closest perfect square above 9680 is 9801 (which is 99^2). √9680 falls somewhere between 98 and 99.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (9680 - 9600) ÷ (9801 - 9600) = 80 ÷ 201 = 0.398

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 98 + 0.398 = 98.398. So the square root of 9680 is approximately 98.398.

• 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 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: Prime factorization is the process of expressing a number as the product of prime numbers.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6 × 6.