Square Root of 2304

The square of root 2304 is 48. The square root of a number is a number that, when multiplied two times to itself, gives the perfect square number. The square root of 2304 is written as √2304, where ‘√’ is called the radical sign. The square root of 2304 in exponential form is written as (2304)^½ or (2304)^0.5 . Since the square root of 2304 is a whole number, hence 2304 is a perfect square number. In this article, we shall learn how to find the square root of 2304.

^.What is the Square Root of 2304?

The square root of 2304 is a number whose square is 2304. Now, if we square both 48 and –48, we get the answer 2304. Thus, the square root of 2304 is both 48 and –48, we choose the positive or the negative root as per the requirement of the problem.

We can also express the square root of 2304 as the roots of the quadratic equation x² – 2304 = 0

x² – 2304 = 0

⇒ x² = 2304 (taking square roots on both sides)

⇒ x = √2304

⇒ x = ± 48

Check out the properties of perfect square numbers.

How to Find the Square Root of 2304?

Let us calculate the square root of 2304 using different methods:

• Repeated subtraction method
• Prime factorisation method
• Long division method

Repeated Subtraction Method

Square of any perfect square number can be determined by the repeated subtraction method. In this method, we have to repeatedly subtract the given number by consecutive odd numbers until we get the answer zero. The nth odd number for which we get the answer zero is the square root of the given number.

Thus, we have 2304 – 1 = 2303, 2303 – 3 = 2300, 2300 – 5 = 2295, continuing in this way we get for the 48th odd number the answer is zero.

∴ 1 + 3 + 5 + 7 + 9 + … + 95 = 2304.

For larger numbers, this method is not preferable. Instead, we can use the prime factorisation method to calculate the square roots.

Prime Factorisation Method

We shall prime factorise the given number and then make pairs of two for each number to find the square root of the number.

Prime factorisation of 2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

The square root of 2304 = √(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 2 × 2 × 3 = 48.

Thus, to find the square root of any number by the prime factorisation method, the following are steps:

• Prime factorise the given number
• Make pairs of two for each of the prime factors.
• Take only one prime factor for each pair.
• If any prime remains unpaired, then the number is not a perfect square.

Long Division Method

To calculate the square root of 2304 by the long division method, we make pairs of digits of 2304 from right to left. Then perform the division as follows:

To learn how to find the square root of any number by the long division method, click here.

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Solved Examples on Square Root of 2304

Example 1:

Find the smallest number which must be multiplied to 384 to make it a perfect square number. Also, find the square root of the perfect square number.

Solution:

Prime factorisation of 384 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3

We observe that 2 and 3 remain unpaired, thus 2 × 3 = 6 must be multiplied to 384.

384 × 6 = 2304 is the perfect square number.

Square root of 2304 = √(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 2 × 2 × 3 = 48.

Example 2:

What is the length of the sides of a square whose area is 2304 cm².

Solution:

Let the length of the side be x.

Then, the area of square = x² = 2304 cm²

⇒ x = √(2304) = 48 cm.

Therefore, the length of the sides of the square is 48 cm.

Example 3:

Find the diameter of the sphere whose total surface area is 28,938.24 cm². (Use 𝜋 = 3.14)

Solution:

Let r be the radius of the sphere.

The total surface area of the sphere = 4𝜋r² = 28,938.24 cm²

⇒ r² = (28,938.24)/( 4 × 3.14) = 2304

⇒ r = √2304 = 48 cm

∴ diameter of the sphere is 2 × 48 = 96 m.