Square Root of 9900
2026-02-28 09:12 Diff
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Last updated on August 5, 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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 9900.

What is the Square Root of 9900?

The square root is the inverse of the square of the number. 9900 is not a perfect square. The square root of 9900 is expressed in both radical and exponential form. In the radical form, it is expressed as √9900, whereas (9900)^(1/2) in the exponential form. √9900 ≈ 99.4987, 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.

Finding the Square Root of 9900

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

Square Root of 9900 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 9900 is broken down into its prime factors.

Step 1: Finding the prime factors of 9900

Breaking it down, we get 2 × 2 × 3 × 3 × 5 × 5 × 11: 2^2 × 3^2 × 5^2 × 11

Step 2: Now we found out the prime factors of 9900. The second step is to make pairs of those prime factors. The square root is found by taking one number from each pair of the same number, which yields: 2 × 3 × 5 × √11 = 30√11

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Square Root of 9900 by Long Division Method

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 9900, we need to group it as 99 and 00.

Step 2: Now we need to find n whose square is less than or equal to 99. We can say n is '9' because 9 × 9 = 81, which is less than 99. Now the quotient is 9, after subtracting 81 from 99, the remainder is 18.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 9 + 9, we get 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 18n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 18n × n ≤ 1800. Let us consider n as 9, now 18 × 9 = 162.

Step 6: Subtract 1620 from 1800; the difference is 180, and the quotient is 99.

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 18000.

Step 8: Now we need to find the new divisor; let's consider 189 because 1890 × 9 = 17010.

Step 9: Subtracting 17010 from 18000, we get the result 990.

Step 10: Now the quotient is 99.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero.

So the square root of √9900 is approximately 99.50.

Square Root of 9900 by Approximation Method

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 9900 using the approximation method.

Step 1: Now we have to find the closest perfect square of √9900. The smallest perfect square less than 9900 is 9801, and the largest perfect square more than 9900 is 10000. √9900 falls somewhere between 99 and 100.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (9900 - 9801) / (10000 - 9801) = 99/199 = 0.4975

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 99 + 0.4975 = 99.4975, so the square root of 9900 is approximately 99.50.

Common Mistakes and How to Avoid Them in the Square Root of 9900

Students do make mistakes while finding the square root, likewise forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √9900?

Okay, lets begin

The area of the square is 9900 square units.

Explanation

The area of the square = side^2.

The side length is given as √9900.

Area of the square = side^2 = √9900 × √9900 = 9900.

Therefore, the area of the square box is 9900 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 9900 square feet is built; if each of the sides is √9900, what will be the square feet of half of the building?

Okay, lets begin

4950 square feet

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 9900 by 2 = we get 4950.

So half of the building measures 4950 square feet.

Well explained 👍

Problem 3

Calculate √9900 × 5.

Okay, lets begin

497.5

Explanation

The first step is to find the square root of 9900, which is approximately 99.50, the second step is to multiply 99.50 with 5.

So 99.50 × 5 = 497.5.

Well explained 👍

Problem 4

What will be the square root of (9800 + 100)?

Okay, lets begin

The square root is approximately 99.50.

Explanation

To find the square root, we need to find the sum of (9800 + 100). 9800 + 100 = 9900, and then √9900 ≈ 99.50.

Therefore, the square root of (9800 + 100) is approximately ±99.50.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √9900 units and the width ‘w’ is 100 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 399 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√9900 + 100) = 2 × (99.50 + 100) = 2 × 199.50 = 399 units.

Well explained 👍

FAQ on Square Root of 9900

1.What is √9900 in its simplest form?

The prime factorization of 9900 is 2 × 2 × 3 × 3 × 5 × 5 × 11, so the simplest form of √9900 = √(2^2 × 3^2 × 5^2 × 11) = 2 × 3 × 5 × √11 = 30√11.

2.Mention the factors of 9900.

Factors of 9900 are 1, 2, 3, 4, 5, 6, 9, 10, 11, 15, 18, 20, 22, 25, 30, 33, 44, 45, 50, 55, 66, 75, 90, 99, 100, 110, 150, 165, 198, 225, 275, 300, 330, 450, 495, 550, 990, 1100, 1485, 1650, 1980, 2475, 2970, 3300, 4950, and 9900.

3.Calculate the square of 9900.

We get the square of 9900 by multiplying the number by itself, that is 9900 × 9900 = 98010000.

4.Is 9900 a prime number?

9900 is not a prime number, as it has more than two factors.

5.9900 is divisible by?

9900 has many factors; those are 1, 2, 3, 4, 5, 6, 9, 10, 11, 15, 18, 20, 22, 25, 30, 33, 44, 45, 50, 55, 66, 75, 90, 99, 100, 110, 150, 165, 198, 225, 275, 300, 330, 450, 495, 550, 990, 1100, 1485, 1650, 1980, 2475, 2970, 3300, 4950, and 9900.

Important Glossaries for the Square Root of 9900

  • 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.
     
  • Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 9900 is 2 × 2 × 3 × 3 × 5 × 5 × 11.
     
  • Perfect square: A perfect square is a number that is the square of an integer. For example, 100 is a perfect square because it is 10^2.
     
  • Exponent: A mathematical notation indicating the number of times a number is multiplied by itself. For example, in 5^3, the exponent is 3, indicating 5 is multiplied by itself three times.

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Jaskaran Singh Saluja

About the Author

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.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.