Square Root of 139.25

The square root is the inverse operation of squaring a number. 139.25 is not a perfect square. The square root of 139.25 can be expressed in both radical and exponential forms. In radical form, it is expressed as √139.25, whereas in exponential form it is (139.25)^(1/2). The square root of 139.25 is approximately 11.799, which is an irrational number because it cannot be expressed as a simple 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, methods such as 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 useful for non-perfect square numbers. Here is how to find the square root using the long division method, step by step:

Step 1: Group the digits of 139.25 from right to left into pairs: 39|25.

Step 2: Find a number whose square is less than or equal to 39. The number is 6 (since 6 x 6 = 36).

Step 3: Subtract 36 from 39, and bring down 25 to get 325.

Step 4: Double the divisor (6) to get 12. Now find a digit n such that 12n * n is close to 325. The digit n is 2 (since 122 x 2 = 244).

Step 5: Subtract 244 from 325 to get 81.

Step 6: Add a decimal point to the quotient, and bring down 00 to get 8100.

Step 7: The new divisor is 124. Find a digit n such that 124n * n is close to 8100. The digit n is 6 (since 1246 x 6 = 7476).

Step 8: Subtract 7476 from 8100 to get 624.

Step 9: Continue the process to get more decimal places.

So, the square root of 139.25 is approximately 11.799.

The approximation method is a simpler way to find the square root of a number:

Step 1: Identify the two perfect squares between which 139.25 lies. The closest perfect squares are 121 (11²) and 144 (12²).

Step 2: √139.25 lies between 11 and 12.

Step 3: Use interpolation to estimate: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) = (139.25 - 121) / (144 - 121)

This results in approximately 0.799. Adding the initial whole number 11 to the decimal point, we get 11 + 0.799 = 11.799.

So, the approximate square root of 139.25 is 11.799.

• Square root: The square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a fraction p/q, where p and q are integers with q ≠ 0. The square root of most non-perfect squares is irrational.
   
• Decimal: A decimal number includes a whole number and a fractional part separated by a decimal point, such as 7.86 or 11.799.
   
• Long Division Method: A step-by-step process to find square roots of non-perfect square numbers by grouping, dividing, and subtracting.
   
• Perfect Square: A perfect square is an integer that is the square of another integer. For example, 144 is a perfect square because it is 12^2.