Square Root of 29.77

The square root is the inverse of the square of the number. 29.77 is not a perfect square. The square root of 29.77 is expressed in both radical and exponential form. In the radical form, it is expressed as √29.77, whereas (29.77)^(1/2) in the exponential form. √29.77 ≈ 5.456, 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 and approximation methods 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 29.77, we consider it as 29.77.

Step 2: Now we need to find a number whose square is close to or less than 29. The number is 5 because 5 x 5 = 25. Now the quotient is 5, and the remainder is 4.

Step 3: Let us bring down 77, making the new dividend 477. Add the old divisor with the same number (5 + 5) to get 10, which will be our new divisor.

Step 4: The new divisor will be 10n, where we need to find the largest digit n such that 10n × n ≤ 477. In this case, n is 4 because 104 x 4 = 416.

Step 5: Subtract 416 from 477, and the remainder is 61. The quotient is now 5.4.

Step 6: Since the remainder is not zero, we need to continue the calculation. Add a decimal point, and bring down two zeroes to make it 6100.

Step 7: The new divisor becomes 108. Using the long division method, find the next digit.

Step 8: Continue these steps until you get a more precise value of the square root to the required decimal places.

The approximate square root of √29.77 is 5.456.

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 29.77 using the approximation method.

Step 1: Now we have to find the closest perfect square to √29.77.

The smallest perfect square less than 29.77 is 25, and the largest perfect square greater than 29.77 is 36.

√29.77 falls somewhere between 5 and 6.

Step 2: Now we need to apply the formula:

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

Using this formula (29.77 - 25) ÷ (36 - 25) = 0.433.

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 5 + 0.433 = 5.433, so the approximate square root of 29.77 is 5.433.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the 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.
   
• Approximation: A method used to find an approximate value of a square root when an exact value is not easily obtainable.
   
• Long Division Method: A step-by-step method used to find the square root of non-perfect squares by dividing numbers in a structured manner.