Square Root of 143.72

The square root is the inverse of the square of the number. 143.72 is not a perfect square. The square root of 143.72 is expressed in both radical and exponential forms. In the radical form, it is expressed as √143.72, whereas (143.72)^(1/2) in the exponential form. √143.72 ≈ 11.9883, 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 the long division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization is not applicable for non-perfect squares like 143.72, as it involves decimals and does not yield a simple expression using integers. For non-perfect squares, we rely on numerical methods such as the long division or approximation method instead.

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. For 143.72, consider it as 14372 (ignoring the decimal temporarily).

Step 2: Find the largest integer n such that n^2 is less than or equal to 143. The integer is 11 because 11^2 = 121, which is less than 143. Subtract 121 from 143 to get a remainder of 22.

Step 3: Bring down 72 to make the dividend 2272. Double the quotient (11) to get 22, which becomes part of our new divisor.

Step 4: Find n such that (220 + n) * n is less than or equal to 2272. Using n = 9, we get (220 + 9) * 9 = 2079.

Step 5: Subtract 2079 from 2272 to get a remainder of 193, and the quotient becomes 11.9.

Step 6: To continue, add decimal places and bring down pairs of zeroes. Repeat the process to reach the desired precision.

The square root of √143.72 is approximately 11.9883.

The approximation method is another method for finding square roots. It is an easy way to estimate the square root of a given number. Now let us learn how to find the square root of 143.72 using the approximation method.

Step 1: Identify the closest perfect squares around 143.72. The closest perfect square below 143.72 is 121 (√121 = 11), and the one above is 144 (√144 = 12).

Step 2: Using linear approximation, estimate where 143.72 falls between these perfect squares.

Step 3: The formula for linear approximation is: (143.72 - 121) / (144 - 121) = (22.72) / 23 ≈ 0.9878

Step 4: Add this result to 11 (the square root of the lower perfect square). Therefore, the approximate square root is 11 + 0.9878 ≈ 11.9878.

So, the square root of 143.72 is approximately 11.9883.

• Square root: A square root is the inverse of a square. For 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; however, it is always the positive square root that is used in real-world applications, known as the principal square root.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A step-by-step process used to divide numbers and find square roots for non-perfect squares, involving grouping and approximating.