Square Root of 3.02

The square root is the inverse of the square of the number. 3.02 is not a perfect square. The square root of 3.02 is expressed in both radical and exponential forms. In the radical form, it is expressed as √3.02, whereas (3.02)^(1/2) in the exponential form. √3.02 ≈ 1.738, 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:

• 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 3.02, we need to consider it as 3.02.

Step 2: Now we need to find n whose square is closest to 3. In this case, n is 1 because 1 × 1 is less than or equal to 3. Now the quotient is 1 after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down the next number 02, making it 202. Add the old divisor with the same number 1 + 1 to 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 such that 2n × n is less than or equal to 202.

Step 5: The next step is finding 2n × n ≤ 202. Let us consider n as 7, now 2 × 7 × 7 = 196.

Step 6: Subtract 196 from 202, the difference is 6, and the quotient is 1.7.

Step 7: Since we have only one decimal place, continue the process by bringing down two zeroes. Now the new dividend is 600.

Step 8: Now, we need to find the new divisor. Using 34 as the new divisor, 34 × 8 = 272.

Step 9: Subtracting 272 from 600 we get the result 328.

Step 10: Now the quotient is 1.73.

Step 11: Continue doing these steps until we get the desired number of decimal places or the remainder becomes zero.

So the square root of √3.02 is approximately 1.738.

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 3.02 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3.02. The smallest perfect square less than 3.02 is 1 and the largest perfect square more than 3.02 is 4. √3.02 falls somewhere between 1 and 2.

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 (3.02 - 1) / (4 - 1) ≈ 0.673. 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 1 + 0.673 ≈ 1.673, so the square root of 3.02 is approximately 1.673.

Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. Now let us look at a few of these mistakes that students tend to make in detail.

• 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, where q is not equal to zero and p and q are integers.

• 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 method used to find the square root of numbers that are not perfect squares by dividing them systematically.

• Approximation Method: A technique used to estimate the square root of numbers by finding two closest perfect squares and using interpolation.