Square Root of 55.68

The square root is the inverse of the square of a number. 55.68 is not a perfect square. The square root of 55.68 is expressed in both radical and exponential form.

In the radical form, it is expressed as √55.68, whereas in the exponential form it is (55.68)^(1/2). √55.68 ≈ 7.458, 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, the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method 
• Approximation method

Since 55.68 is not a perfect square, the prime factorization method is not suitable for finding its square root accurately. Instead, methods like long division and approximation are preferred.

The long division method is 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, group the numbers from right to left. In the case of 55.68, group it as 55 and 68.

Step 2: Now find n whose square is less than or equal to 55. We can say n as ‘7’ because 7 × 7 = 49, which is less than 55. The quotient is 7; after subtracting 49 from 55, the remainder is 6.

Step 3: Bring down 68, making the new dividend 668. Double the quotient and use it as a new divisor. Hence, 2 × 7 = 14.

Step 4: Find a digit x such that 14x × x ≤ 668. The suitable value is x = 4, so 144 × 4 = 576.

Step 5: Subtract 576 from 668, and the remainder is 92. The quotient is now 7.4.

Step 6: Add a decimal point and bring down two zeros, making the dividend 9200.

Step 7: Double the quotient considering the decimal, making it 148. Find a digit y such that 148y × y ≤ 9200. The suitable value is y = 6, so 1486 × 6 = 8916.

Step 8: Subtract 8916 from 9200, and the remainder is 284.

Step 9: Continue these steps to get two numbers after the decimal point. If there is no remainder, continue until the remainder is zero.

So the square root of √55.68 ≈ 7.46.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 55.68 using the approximation method.

Step 1: Identify the closest perfect squares around 55.68. The smallest perfect square less than 55.68 is 49, and the largest perfect square more than 55.68 is 64. √55.68 falls between 7 and 8.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (55.68 - 49) ÷ (64 - 49) = 6.68 / 15 ≈ 0.445. Add this to the square root of 49, which is 7. Therefore, the approximate square root of 55.68 is 7 + 0.445 = 7.445.

Thus, √55.68 ≈ 7.45.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction of two integers. Example: √2 is irrational.
   
• Decimal: A decimal number includes a whole number and a fractional part separated by a decimal point. Example: 7.46.
   
• Approximation: The process of finding a value that is close enough to the correct value, usually within a certain margin of error.
   
• Long division: A method used to divide large numbers by breaking them into easier steps, often used for finding square roots of non-perfect squares.