Square Root of 232

The square root is the inverse of the square of a number. 232 is not a perfect square. The square root of 232 is expressed in both radical and exponential form. In the radical form, it is expressed as √232, whereas (232)^(1/2) in the exponential form. √232 ≈ 15.2315, 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 method and approximation method 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 232 is broken down into its prime factors.

Step 1: Finding the prime factors of 232. Breaking it down, we get 2 × 2 × 2 × 29: 2³ × 29¹

Step 2: Now we found out the prime factors of 232. The second step is to make pairs of those prime factors. Since 232 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 232 using prime factorization is not straightforward.

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 232, we need to group it as 32 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n as ‘1’ because 1 × 1 is less than or equal to 2. Now the quotient is 1, after subtracting 2 - 1 the remainder is 1.

Step 3: Now let us bring down 32 which is the new dividend. Add the old divisor with the same number 1 + 1 we get 2 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 132. Let us consider n as 5, now 2 × 5 × 5 = 125.

Step 6: Subtract 132 from 125, the difference is 7, and the quotient is 15.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 700.

Step 8: Now we need to find the new divisor. Consider the new divisor as 152, because 152 × 4 = 608.

Step 9: Subtracting 608 from 700 we get the result 92.

Step 10: Now the quotient is 15.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values continue till the remainder is zero. So the square root of √232 ≈ 15.23.

The 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 232 using the approximation method.

Step 1: Now we have to find the closest perfect square of √232. The smallest perfect square less than 232 is 225 and the largest perfect square greater than 232 is 256. √232 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (232 - 225) ÷ (256 - 225) = 7/31 ≈ 0.226. Using the formula we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 15 + 0.226 ≈ 15.23. Therefore, the square root of 232 is approximately 15.23.

• Square root: A square root is the inverse of a square. Example: 4² = 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, 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 a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: The process of breaking down a number into its prime factors.
   
• 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.