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.

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

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.

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.

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.

• 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.