Square Root of 30.77

The square root is the inverse of the square of the number. 30.77 is not a perfect square. The square root of 30.77 is expressed in both radical and exponential form. In the radical form, it is expressed as √30.77, whereas (30.77)^(1/2) in the exponential form. √30.77 ≈ 5.547, 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 prime factorization method involves expressing a number as a product of its prime factors. However, 30.77 is not a whole number, and thus, the prime factorization method is not applicable directly. We use other methods for non-perfect squares and decimals like 30.77.

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 30.77, consider the whole number and decimal part separately.

Step 2: Find the largest number whose square is less than or equal to 30. The number is 5 because 5 × 5 = 25. The quotient is 5.

Step 3: Subtract 25 from 30 to get 5. Bring down the next pair of digits from the decimal part, making it 577.

Step 4: Double the quotient (5) and use it as the new divisor's first part (10). Determine the digit x such that 10x × x is less than or equal to 577.

Step 5: The number x is 5 because 105 × 5 = 525, which is less than 577.

Step 6: Subtract 525 from 577 to get 52. The quotient is now 5.5.

Step 7: Since we need more precision, continue the process by adding decimal places and bringing down pairs of zeros.

Following these steps will yield a square root of approximately 5.547 for √30.77.

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 30.77 using the approximation method.

Step 1: Identify the closest perfect squares around 30.77.

The closest perfect squares are 25 (5²) and 36 (6²). √30.77 falls between 5 and 6.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (30.77 - 25) ÷ (36 - 25) ≈ 0.525.

Step 3: Add the value obtained to the smaller integer: 5 + 0.525 ≈ 5.525.

Thus, using the approximation method, the square root of 30.77 is approximately 5.547.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, so √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.
   
• Approximation: The method of finding a value that is close enough to the right answer, typically with some degree of error, is called approximation.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing, multiplying, and subtracting in a systematic manner.