Square Root of 37

The square root of 37 can be determined using different methods, which we shall discuss in detail. We get an irrational root because 37 is not a perfect square number. A perfect square number is a number that can be written as a square of an integer and whose square root is a whole number. The square root of a number is just the opposite of squaring. In radical form, the square root of 37 is denoted as √37, and in the exponential form, it is written as (37)^½. Let us learn how to find the square root of 37.

Learn more about square and square roots.

What Is the Square Root of 37?

The square root of 37 is a number whose square gives the result of 37. Now, 37 is not a perfect square number because we cannot find any integer which could be multiplied twice to get 37. Thus, we find an approximate value of the square root of 37, as it is an irrational number.

The square root of 37 can also be determined by finding the roots of the quadratic equation

x² – 37 = 0

⇒ x² = 37

Taking square roots on both sides, we get,

⇒ x = √37

⇒ x = ± 6.0827.

How to Find the Square Root of 37?

Let us find the square root of 37 using Newton’s square root equation, prime factorisation and long division method. The repeated subtraction method will not work as 37 is not a perfect square, and its square root is an irrational number.

Newton’s Square Root Equation

Newton’s square root equation to find the square root of any number is given by

√N ≈ ½ (N/A + A), where

N is the number whose square root we need to find.

A is the approximate guess square root number.

Now, we have N = 37, as 6² = 36 < 37, we take A = 6. Let us put all the values in the formula to find the square root of 37.

√37 ≈ ½ (37/6 + 6) = ½ (6.1667 + 6) ≈ 6.08

Prime Factorisation Method

To find the square root of 37 by the prime factorisation method, we first prime factorise 37 and then make pairs of two to get the square root. Since 37 is a prime number,

Prime factorisation of 37 = 37 × 1

Square root of 37 = √[37 × 1] = √37

Long Division Method

To find the square root of 37 by the long division method, we shall write 37 as the dividend and pair its digits from right to left. Now, we shall calculate the square root:

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 37

Example 1:

Find the perimeter of the circle whose area is 37𝜋 m².

Solution:

Let r be the radius of the circle.

The area of the circle = 𝜋r² = 37𝜋 m²

⇒ r² = 37 (taking square root on both sides)

⇒ r = √37

⇒ r ≈ 6.08 m (taking the positive root as length cannot be negative)

∴ the perimeter of the circle = 2 × 𝜋 × 6.08 = 38.1824 m

Example 2:

Find the length of the base of a triangle whose area is 3.7 m² and the height is one-fifth of its base.

Solution:

Let b be the base of the triangle.

Height of the triangle = b/5

Area of the triangle = ½ × b × b/5 = 3.7 m²

⇒ 1/10 × b² = 3.7

⇒ b² = 3.7 × 10 = 37 (taking square root on both the sides)

⇒ b = √37 ≈ 6.08 m

∴ the length of the base of the triangle is 6.08 m.

Example 3:

What is the smallest number that should be subtracted from 37 to make it a perfect square number? Also, find the square root of that number obtained.

Solution:

Clearly, 6² = 36 < 37

Therefore, we must subtract 1 from 37 to make it a perfect square number.

√36 = ± 6