Square Root of 37.5
2026-02-28 01:10 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 37.5

What is the Square Root of 37.5?

The square root is the inverse of the square of the number. 37.5 is not a perfect square. The square root of 37.5 is expressed in both radical and exponential forms. In the radical form, it is expressed as √37.5, whereas (37.5)^(1/2) in the exponential form. √37.5 ≈ 6.12372, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 37.5

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 37.5 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 37.5 is broken down into its prime factors:

Step 1: Convert 37.5 into a fraction: 375/10.

Step 2: Find the prime factors of the numerator 375, which are 3 × 5 × 5 × 5.

Step 3: Find the prime factors of the denominator 10, which are 2 × 5.

Step 4: Therefore, √37.5 = √(375/10) = √(3 × 5 × 5 × 5/2 × 5).

Step 5: Simplify the expression: √(3 × 5 × 5 × 5/2 × 5) = √(3 × 5 × 5/2).

Since 37.5 is not a perfect square, calculating using prime factorization will still require approximation.

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Square Root of 37.5 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now 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 37.5, consider it similar to 3750 and group it as 37 and 50.

Step 2: Now we need to find n whose square is less than or equal to 37. We can say n as ‘6’ because 6 × 6 = 36, which is lesser than or equal to 37. Now the quotient is 6 and the remainder is 1.

Step 3: Bring down 50, making the new dividend 150.

Step 4: The new divisor is 2 × 6 = 12. Find n such that 12n × n ≤ 150.

Step 5: Consider n as 1, now 121 × 1 = 121.

Step 6: Subtract 121 from 150, the remainder is 29, and the quotient is 61.

Step 7: Since the dividend is less than the divisor, add a decimal point and two zeros to the dividend, making it 2900.

Step 8: Find the new divisor which is 122 because 1220 × 2 = 2440.

Step 9: Subtract 2440 from 2900, resulting in 460. Step 10: The quotient is 6.12.

Step 11: Continue doing these steps until you achieve the desired decimal places.

So the square root of √37.5 ≈ 6.12372.

Square Root of 37.5 by Approximation Method

Approximation method is another method for finding the square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 37.5 using the approximation method.

Step 1: Find the closest perfect squares to √37.5. The smallest perfect square less than 37.5 is 36 and the largest perfect square greater than 37.5 is 49. √37.5 falls between 6 and 7.

Step 2: Apply the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula: (37.5 - 36) / (49 - 36) = 1.5 / 13 ≈ 0.1154

 Step 3: Add this decimal to the smaller perfect square root: 6 + 0.1154 ≈ 6.1154.

So, the approximate square root of 37.5 is 6.1154.

Common Mistakes and How to Avoid Them in the Square Root of 37.5

Students make mistakes while finding the square root, like forgetting about the negative square root or skipping long division steps. Now let us look at a few of those mistakes that students tend to make in detail.

Problem 1

Can you help Max find the area of a square box if its side length is given as √37.5?

Okay, lets begin

The area of the square is approximately 37.5 square units.

Explanation

The area of the square = side^2.

The side length is given as √37.5.

Area of the square = side^2 = √37.5 × √37.5 ≈ 6.12372 × 6.12372 = 37.5.

Therefore, the area of the square box is 37.5 square units.

Well explained 👍

Problem 2

A square-shaped garden measuring 37.5 square meters is planned; if each of the sides is √37.5, what will be the square meters of half of the garden?

Okay, lets begin

18.75 square meters

Explanation

We can just divide the given area by 2 as the garden is square-shaped.

Dividing 37.5 by 2 gives us 18.75.

So half of the garden measures 18.75 square meters.

Well explained 👍

Problem 3

Calculate √37.5 × 5.

Okay, lets begin

Approximately 30.6186

Explanation

The first step is to find the square root of 37.5, which is approximately 6.12372.

The second step is to multiply 6.12372 with 5. So 6.12372 × 5 ≈ 30.6186.

Well explained 👍

Problem 4

What will be the square root of (25 + 12.5)?

Okay, lets begin

The square root is 6.

Explanation

To find the square root, we need to find the sum of (25 + 12.5). 25 + 12.5 = 37.5, and then √37.5 ≈ 6.12372.

Therefore, the square root of (25 + 12.5) is approximately 6.12372.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √37.5 units and the width ‘w’ is 10 units.

Okay, lets begin

The perimeter of the rectangle is approximately 32.24744 units.

Explanation

Perimeter of the rectangle = 2 × (length + width)

Perimeter = 2 × (√37.5 + 10) ≈ 2 × (6.12372 + 10) ≈ 2 × 16.12372 ≈ 32.24744 units.

Well explained 👍

FAQ on Square Root of 37.5

1.What is √37.5 in its simplest form?

The simplest form of √37.5 is obtained by converting it into a fraction and simplifying: √(375/10) = √(3 × 5 × 5/2).

2.Mention the factors of 37.5.

Factors of 37.5 are 1, 2.5, 5, 7.5, 12.5, 15, and 37.5.

3.Calculate the square of 37.5.

We get the square of 37.5 by multiplying the number by itself: 37.5 × 37.5 = 1406.25.

4.Is 37.5 a prime number?

37.5 is not a prime number, as it is not an integer, and prime numbers are defined only for integers.

5.37.5 is divisible by?

37.5 is divisible by 1, 2.5, 5, 7.5, 12.5, 15, and 37.5.

Important Glossaries for the Square Root of 37.5

  • Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
     
  • Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
     
  • Principal square root: A number has both positive and negative square roots, but it is usually the positive square root that is used due to its practical applications. This is known as the principal square root.
     
  • Approximation method: A method used to estimate the value of a square root by finding the nearest perfect squares and using them to calculate a closer value.
     
  • Long division method: A method used to find the square root of a number by dividing it into groups and performing a series of divisions and subtractions to achieve the desired precision.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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