Square Root of 3.3

The square root is the inverse of the square of the number. 3.3 is not a perfect square. The square root of 3.3 is expressed in both radical and exponential form. In the radical form, it is expressed as √3.3, whereas (3.3)^(1/2) in the exponential form. √3.3 ≈ 1.81659, 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

• Prime Factorization Method
• Long division method 
• Approximation method

The product of prime factors is the prime factorization of a number. However, 3.3 is not an integer and cannot be broken down into prime factors like whole numbers. Therefore, calculating the square root of 3.3 using prime factorization is not applicable. We proceed with the long division or approximation methods.

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. Here, 3.3 can be treated as 330 when considering two decimal places.

Step 2: Start with 1 as a potential quotient because 1² = 1, which is less than 3. Subtract 1 from 3 to get a remainder of 2.

Step 3: Bring down 30, making the new dividend 230.

Step 4: The next step is finding 2n × n such that it is less than or equal to 230. If n = 1, then 21 × 1 = 21.

Step 5: Subtract 21 from 230, resulting in a remainder of 209.

Step 6: Bring down the next pair of zeroes. Repeat the steps to refine the quotient. Continue these steps until you get the desired precision.

The process gives the square root as approximately 1.8165.

The approximation method is another method for finding square roots, and 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 3.3 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √3.3. The smallest perfect square less than 3.3 is 1, and the nearest perfect square greater than 3.3 is 4. √3.3 falls somewhere between 1 and 2.

Step 2: Now we need to apply the formula: (3.3 - 1) / (4 - 1) = 0.7667 Using the formula, we identified the decimal point of our square root. The next step is adding the base value to the decimal number, which is 1 + 0.7667 ≈ 1.7667. So, the square root of 3.3 is approximately 1.8165.

• 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 p and q are integers and q ≠ 0.

• Rational number: A rational number is a number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

• 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.

• Approximation: Approximation involves finding a value that is close to the actual value but not exact, often used when dealing with irrational numbers or complex calculations.