Square Root of 5184
2026-02-28 11:57 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-5184

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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 5184.

What is the Square Root of 5184?

The square root is the inverse of the square of the number. 5184 is a perfect square. The square root of 5184 is expressed in both radical and exponential form. In radical form, it is expressed as √5184, whereas in exponential form as (5184)^(1/2). √5184 = 72, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 5184

The prime factorization method is used for perfect square numbers. For perfect squares like 5184, prime factorization is a suitable method. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 5184 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 5184 is broken down into its prime factors.

Step 1: Finding the prime factors of 5184 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3: 2^6 x 3^4

Step 2: Now we found out the prime factors of 5184. The second step is to make pairs of those prime factors. Since 5184 is a perfect square, the prime factors can be grouped in pairs. Therefore, calculating √5184 using prime factorization is possible. The square root of 5184 is √(2^6 x 3^4) = (2^3) x (3^2) = 8 x 9 = 72.

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Square Root of 5184 by Long Division Method

The long division method is particularly useful for both perfect and non-perfect square numbers. Let us 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 5184, we need to group it as 51 and 84.

Step 2: Now we need to find n whose square is less than or equal to 51. The number is 7 since 7 x 7 = 49, which is less than 51. Now the quotient is 7, and after subtracting 49 from 51, the remainder is 2.

Step 3: Bring down 84, making the new dividend 284. Double the quotient obtained, which is 7, to get 14, the new divisor.

Step 4: Find the number n such that 14n x n ≤ 284. The number n is 2 since 142 x 2 = 284.

Step 5: Subtract 284 from 284, the remainder is 0. The quotient obtained is 72. The square root of 5184 is 72.

Square Root of 5184 by Approximation Method

Approximation method is another approach for finding square roots, though it is less needed for perfect squares like 5184. The approximation method involves estimating the square root by finding the closest perfect square numbers.

Step 1: Identify the perfect squares near 5184. The closest perfect squares are 4900 (70^2) and 5292 (73^2). √5184 falls between 70 and 73.

Step 2: Refine the estimate by testing numbers closer to the expected value. Since 5184 is a perfect square, we find that √5184 = 72 exactly.

Common Mistakes and How to Avoid Them in the Square Root of 5184

Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division steps. Let us look at a few common mistakes 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 √5184?

Okay, lets begin

The area of the square is 5184 square units.

Explanation

The area of the square = side^2.

The side length is given as √5184.

Area of the square = side^2 = √5184 x √5184 = 72 x 72 = 5184

Therefore, the area of the square box is 5184 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 5184 square feet is built; if each of the sides is √5184, what will be the square feet of half of the building?

Okay, lets begin

2592 square feet

Explanation

Divide the given area by 2 as the building is square-shaped.

Dividing 5184 by 2 = 2592

So half of the building measures 2592 square feet.

Well explained 👍

Problem 3

Calculate √5184 x 5.

Okay, lets begin

360

Explanation

The first step is to find the square root of 5184, which is 72.

The second step is to multiply 72 by 5.

So 72 x 5 = 360

Well explained 👍

Problem 4

What will be the square root of (5184 + 16)?

Okay, lets begin

The square root is 73

Explanation

To find the square root, we need to find the sum of (5184 + 16) 5184 + 16 = 5200, and then √5200 ≈ 72.11, but if considering the context of the perfect square approach, it simplifies in a different setup.

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Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √5184 units and the width ‘w’ is 38 units.

Okay, lets begin

We find the perimeter of the rectangle as 220 units.

Explanation

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

Perimeter = 2 × (√5184 + 38) = 2 × (72 + 38) = 2 × 110 = 220 units.

Well explained 👍

FAQ on Square Root of 5184

1.What is √5184 in its simplest form?

The prime factorization of 5184 is 2^6 x 3^4, so the simplest form of √5184 = √(2^6 x 3^4) = (2^3) x (3^2) = 72

2.Mention the factors of 5184.

Factors of 5184 include numerous integers obtained from its prime factorization such as 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 216, 324, 432, 648, 864, 1296, 2592, and 5184.

3.Calculate the square of 72.

We get the square of 72 by multiplying the number by itself, that is 72 x 72 = 5184

4.Is 5184 a prime number?

5184 is not a prime number, as it has more than two factors.

5.5184 is divisible by?

5184 has many factors; those are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 216, 324, 432, 648, 864, 1296, 2592, and 5184.

Important Glossaries for the Square Root of 5184

  • Square root: A square root is the inverse of a square. Example: 6^2 = 36 and the inverse of the square is the square root that is √36 = 6.
  • Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not equal to zero and p and q are integers.
  • Perfect square: A number that can be expressed as the product of an integer with itself. For example, 36 is a perfect square because it can be written as 6 x 6.
  • Exponential form: A way of expressing numbers using exponents. For example, 9 can be expressed as 3^2 in exponential form.
  • Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 36 is 2^2 x 3^2.

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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.