Square Root of 18000
2026-02-28 12:51 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 18000.

What is the Square Root of 18000?

The square root is the inverse of the square of the number. 18000 is not a perfect square. The square root of 18000 is expressed in both radical and exponential form. In the radical form, it is expressed as √18000, whereas (18000)(1/2) in the exponential form. √18000 ≈ 134.16408, 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 18000

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods are used. Let us now learn the following methods:

  1. Prime factorization method
  2. Long division method
  3. Approximation method

Square Root of 18000 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 18000 is broken down into its prime factors.

Step 1: Finding the prime factors of 18000 Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5: 23 x 32 x 54

Step 2: Now we found out the prime factors of 18000. The next step is to make pairs of those prime factors. Since 18000 is not a perfect square, therefore the digits of the number can’t be grouped in a complete pair.

Therefore, calculating 18000 using prime factorization is not straightforward.

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Square Root of 18000 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 18000, we need to group it as 00, 80, and 18.

Step 2: Now we need to find n whose square is less than or equal to 18. We can say n is ‘4’ because 4 x 4 = 16, which is less than 18. Now the quotient is 4 after subtracting 16 from 18, the remainder is 2.

Step 3: Now let us bring down 80 which is the new dividend. Add the old divisor with the same number 4 + 4 we get 8, making it our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 280. Let us consider n as 3, now 83 x 3 = 249.

Step 6: Subtract 249 from 280, the difference is 31, and the quotient is 43.

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 zeros to the dividend. Now the new dividend is 3100.

Step 8: Now we need to find the new divisor that is 866 because 866 x 3 = 2598

Step 9: Subtracting 2598 from 3100, we get the result 502.

Step 10: Now the quotient is 134.1

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 √18000 is approximately 134.16

Square Root of 18000 by Approximation Method

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 18000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √18000. The smallest perfect square less than 18000 is 17689, and the largest perfect square greater than 18000 is 18225. √18000 falls somewhere between 133 and 135.

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 (18000 - 17689) ÷ (18225-17689) ≈ 0.16408. 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 134 + 0.16408 ≈ 134.16408, so the square root of 18000 is approximately 134.16408.

Common Mistakes and How to Avoid Them in the Square Root of 18000

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 √18500?

Okay, lets begin

The area of the square is approximately 184.39 square units.

Explanation

The area of the square = side2.

The side length is given as √18500.

Area of the square = side2 = √18500 x √18500 ≈ 136.019 × 136.019 ≈ 184.39.

Therefore, the area of the square box is approximately 184.39 square units.

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Problem 2

A square-shaped building measuring 18000 square feet is built; if each of the sides is √18000, what will be the square feet of half of the building?

Okay, lets begin

9000 square feet

Explanation

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

Dividing 18000 by 2 = we get 9000.

So half of the building measures 9000 square feet.

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Problem 3

Calculate √18000 x 5.

Okay, lets begin

670.82

Explanation

The first step is to find the square root of 18000 which is approximately 134.16.

The second step is to multiply 134.16 with 5.

So 134.16 x 5 ≈ 670.82.

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Problem 4

What will be the square root of (18000 + 400)?

Okay, lets begin

The square root is approximately 134.536.

Explanation

To find the square root, we need to find the sum of (18000 + 400). 18000 + 400 = 18400, and then √18400 ≈ 134.536.

Therefore, the square root of (18000 + 400) is approximately ±134.536.

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Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √18000 units and the width ‘w’ is 50 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 368.32 units.

Explanation

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

Perimeter = 2 × (√18000 + 50) ≈ 2 × (134.16 + 50) ≈ 2 × 184.16 ≈ 368.32 units.

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FAQ on Square Root of 18000

1.What is √18000 in its simplest form?

The prime factorization of 18000 is 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5. Therefore, the simplest form of √18000 = √(23 x 32 x 54).

2.Mention the factors of 18000.

Factors of 18000 include 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 125, 150, 180, 225, 250, 300, 375, 450, 500, 600, 750, 900, 1125, 1500, 1800, 2250, 3000, 4500, 6000, 9000, and 18000.

3.Calculate the square of 18000.

We get the square of 18000 by multiplying the number by itself, that is 18000 x 18000 = 324000000.

4.Is 18000 a prime number?

18000 is not a prime number, as it has more than two factors.

5.18000 is divisible by?

18000 has many factors; it is divisible by 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 125, 150, 180, 225, 250, 300, 375, 450, 500, 600, 750, 900, 1125, 1500, 1800, 2250, 3000, 4500, 6000, 9000, and 18000.

Important Glossaries for the Square Root of 18000

  • Square root: A square root is the inverse of a square. Example: 42 = 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 usually the positive square root that is used in real-world applications. This is known as the principal square root.
  • Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 18000 is 23 x 32 x 54.
  • Decimal: A decimal is a way of representing numbers that have a whole number part and a fractional part, separated by a decimal point. Example: 134.16 is a decimal.

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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.

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