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.

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

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.

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.

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.

• 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.