The square root of 1600 is 40. 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 1600 is written as √1600, where ‘√’ is called the radical sign. The square root of 1600 in exponential form is written as (1600)^½ . Since the square root of 1600 is a whole number, 1600 is a perfect square number. In this article, we shall learn how to find the square root of 1600

What Is the Square Root of 1600?

The square root of 1600 is a number whose square is 1600. Now, if we square both 40 and –40, we get the answer 1600. Thus, the square root of 1600 is both 40 and –40; we choose the positive or the negative root as per the requirement of the problem.

We can also express the square root of 1600 as the roots of the quadratic equation x² – 1600 = 0.

x² – 1600 = 0

⇒ x² = 1600 (taking square roots on both sides)

⇒ x = √(1600)

⇒ x = ± 40

Check out the properties of perfect square numbers.

How to Find the Square Root of 1600?

Let us calculate the square root of 1600 using different methods:

• Repeated subtraction method
• Prime factorisation method
• Long division 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 1600 will be n.

In the 40th step, the answer is zero; hence, the square root of 1600 is 40. This method of finding the square root is simple, but for bigger numbers, it becomes really difficult to subtract successively.

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 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

√1600 = √[(2 × 2) × (2 × 2) × (2 × 2) × (5 × 5)] = 2 × 2 × 2 × 5 = 40

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 1600 by the long division method, we make pairs of digits of 1600 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 1600

Example 1:

Find the smallest number, which must be multiplied by 160 to make it a perfect square number. Also, find the square root of the perfect square number.

Solution:

Prime factorisation of 160 = (2 × 2) × (2 × 2) × 2 × 5

We observe that 2 and 5 remain unpaired. Thus, 2 × 5 = 10 must be multiplied by 160.

∴ 160 × 10 = 1600 is the perfect square number.

Square root of 1600 = √[(2 × 2) × (2 × 2) × (2 × 2) × (5 × 5] = 2 × 2 × 2 × 5 = 40

Example 2:

There are 1600 chairs in an auditorium hall. If there are equal numbers of rows and columns of chairs, find out how many rows there are in the hall.

Solution:

Let there be x numbers of rows and columns in the hall.

Total number of chairs in the hall = x × x = 1600

⇒ x² = 1600

Taking square roots on both sides, we get,

x = √1600 = 40

∴ there are 40 rows of chairs in the hall.

Example 3:

Find the diameter of the sphere whose total surface area is 20,096 cm² (Use 𝜋 = 3.14).

Solution:

Let r be the radius of the sphere.

The total surface area of the sphere = 4𝜋r² = 20,096 cm²

⇒ r² = (20,096)/( 4 × 3.14) = 1600

⇒ r = √1600 = 40 cm

∴ diameter of the sphere is 2 × 40 = 80 cm