Square Root of 37.96

The square root is the inverse of the square of the number. 37.96 is not a perfect square. The square root of 37.96 is expressed in both radical and exponential form.

In the radical form, it is expressed as √37.96, whereas (37.96)^(1/2) in the exponential form. √37.96 = 6.16, which is a rational number because it can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation 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 37.96, we need to group it as 37 and 96.

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

Step 3: Now let us bring down 96, making the new dividend 196. Add the old divisor with the same number 6 + 6 = 12, which will be our new divisor base.

Step 4: We need to find a digit, say n, which satisfies 12n x n ≤ 196. In this case, n is 1 because 121 x 1 = 121. Step 5: Subtract 121 from 196, the difference is 75, and the new quotient is 6.1.

Step 6: Since the dividend is less than the divisor, add a decimal point and two zeros, making the new dividend 7500.

Step 7: Now we need to find the new divisor. The new divisor becomes 122, and finding a suitable n, we choose 6 because 1226 x 6 = 7356.

Step 8: Subtracting 7356 from 7500 gives us 144.

Step 9: Continue doing these steps until we achieve the desired accuracy.

The process shows that √37.96 is approximately 6.16.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 37.96 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 37.96.

The smallest perfect square less than 37.96 is 36, and the largest perfect square greater than 37.96 is 49. √37.96 falls between 6 and 7.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula: (37.96 - 36) / (49 - 36) = 1.96 / 13 ≈ 0.15 Adding this approximation to 6, we get 6 + 0.15 = 6.15.

Thus, the square root of 37.96 is approximately 6.16.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.
  
   
• Rational number: A rational number is a number that can be expressed as a fraction p/q, where q is not equal to zero, and p and q are integers.
  
   
• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 9 is 3.
  
   
• Long division method: A method used to find the square root of non-perfect squares by dividing numbers into pairs and approximating the result.
  
   
• Decimal: A number that represents a fraction in a base-ten system, consisting of a whole number and a fractional part separated by a decimal point (e.g., 6.16).