Square Root of 300

The square root of 300 is an irrational number. The square root of a number is a number that, when multiplied by itself, gives the perfect square number. The square root of 300 in radical form is written as √300, where ‘√’ is called the radical sign. The square root of 300 in exponential form is written as (300)^½ or (300)^0.5 . Since the square root of 300 is not a whole number, 300 is not a perfect square number. In this article, we shall learn how to find the square root of 300.

What Is the Square Root of 300?

The square root of 300 is a number whose square is 300. Let x be a real number, such that

x × x = x² = 300. Now, we have to determine the value of x.

Thus, we can also express the square root of 300 as the roots of the quadratic equation x² – 300 = 0

x² – 300 = 0

⇒ x² = 300 (taking square roots on both sides)

⇒ x = √(300)

⇒ x = ± 10√3

We can verify that by squaring ± 10√3. Thus, we get 300.

Check out the properties of perfect square numbers.

How to Find the Square Root of 300?

Let us calculate the square root of 300 using different methods:

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

Repeated Subtraction Method

To find the square root using this method, we shall successively subtract odd numbers from the given number until we get zero. The nth odd number for which we get the result zero, the square root of 300 will be n.

In the 18th step, we get a negative integer instead of zero. Hence, the square root of 300 cannot be calculated using this method. We can estimate that root 300 is an irrational number between 17 and 18.

Prime Factorisation Method

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

Prime factorisation of 300 = 2 × 2 × 3 × 5 × 5

√300 = √[(2 × 2) × 3 × (5 × 5)] = 2 × 5 × √3 = 10√3

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 300 by the long division method, we make pairs of digits of 300 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.

Approximation Method

We shall use Newton’s formula to find an approximate value of the square root of any number, perfect square or not. The formula is given as

Where,

X = Number whose square root needs to be calculated

A = Guess value of the square root of the given number; we generally take the number whose square is nearest to X.

Here, X = 300, A = 17 as 17² = 289 < 300. Substituting these values in the formula, we get

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Solved Examples on Square Root of 300

Example 1:

Find the smallest number, which must be multiplied by 300 to make it a perfect square number. Also, find the square root of the perfect square number.

Solution:

Prime factorisation of 300 = (2 × 2) × 3 × (5 × 5)

We observe that only 3 remain unpaired. Thus, 3 must be multiplied by 300.

∴ 300 × 3 = 900 is the perfect square number.

Square root of 900 = √[(2 × 2) × (3 × 3) × (5 × 5)] = 2 × 3 × 5 = 30

Example 2:

There are 300 chairs in an auditorium hall. They are arranged in equal numbers of rows and columns of chairs. Find out how many chairs will be left after doing so.

Solution:

The greatest perfect square number less than 300 is 289, which is the square of 17.

Thus, there will be 17 rows and columns of chairs.

The number of chairs left = 300 – 289 = 11

Example 3:

Find the diameter of the sphere whose total surface area is 3768 mm² (Use 𝜋 = 3.14).

Solution:

Let r be the radius of the sphere.

The total surface area of the sphere = 4𝜋r² = 3768 mm²

⇒ r² = (3768)/( 4 × 3.14) = 300

⇒ r = √300 ≈ 17.32 cm

∴ diameter of the sphere is 2 × 17.32 = 34.64 cm