Square Root of 332

The square root is the inverse of the square of the number. 332 is not a perfect square. The square root of 332 is expressed in both radical and exponential form. In radical form, it is expressed as √332, whereas in exponential form it is expressed as (332)^(1/2). √332 ≈ 18.2209, 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, for non-perfect square numbers, we use the long division method and approximation method. 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 332 is broken down into its prime factors.

Step 1: Finding the prime factors of 332 Breaking it down, we get 2 x 2 x 83: 2^2 x 83

Step 2: Now we found out the prime factors of 332. The second step is to make pairs of those prime factors. Since 332 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √332 using prime factorization is not straightforward and will 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 332, we need to group it as 32 and 3.

Step 2: Now we need to find n whose square is <= 3. We can say n is ‘1’ because 1 x 1 = 1, which is less than 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 32, which is the new dividend. Add the old divisor with the same number: 1 + 1 = 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 ≤ 232. Let us consider n as 9, now 2 x 9 x 9 = 162.

Step 6: Subtract 162 from 232, the difference is 70, and the quotient is 19.

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 7000.

Step 8: Now we need to find the new divisor that is 389 because 389 x 9 = 3501.

Step 9: Subtracting 3501 from 7000 we get the result 3499.

Step 10: Now the quotient is 18.9.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero.

So the square root of √332 ≈ 18.22

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 332 using the approximation method.

Step 1: Now we have to find the closest perfect square of √332. The smallest perfect square less than 332 is 324 and the largest perfect square greater than 332 is 361. √332 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula to find the approximate value: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula: (332 - 324) ÷ (361 - 324) ≈ 0.2162 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 18 + 0.2162 ≈ 18.22.

So, the square root of 332 is approximately 18.22.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, so the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, such as √2 or π.

• Principal square root: The principal square root is the positive square root of a number. For example, the principal square root of 9 is 3, not -3.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups of digits and iteratively finding the square root to a desired accuracy.