Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as engineering, finance, etc. Here, we will discuss the 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:
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.
Students do make mistakes while finding square roots, such as forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of the mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √9680?
The area of the square is 936640 square units.
The area of the square = side².
The side length is given as √9680.
Area of the square = side² = √9680 × √9680 = 98.39 × 98.39 ≈ 936640.
Therefore, the area of the square box is approximately 936640 square units.
A square-shaped building measuring 9680 square feet is built; if each of the sides is √9680, what will be the square feet of half of the building?
4840 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 9680 by 2, we get 4840.
So half of the building measures 4840 square feet.
Calculate √9680 × 5.
491.95
The first step is to find the square root of 9680, which is approximately 98.39.
The second step is to multiply 98.39 by 5.
So 98.39 × 5 = 491.95.
What will be the square root of (9500 + 180)?
The square root is approximately 98.39.
To find the square root, we need to find the sum of (9500 + 180). 9500 + 180 = 9680, and then the square root of 9680 is approximately 98.39.
Therefore, the square root of (9500 + 180) is approximately ±98.39.
Find the perimeter of the rectangle if its length ‘l’ is √9680 units and the width ‘w’ is 38 units.
The perimeter is approximately 272.78 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√9680 + 38) = 2 × (98.39 + 38) = 2 × 136.39 = 272.78 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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