Square Root of 55

The square root is the inverse of the square of the number. 55 is not a perfect square. The square root of 55 is expressed in both radical and exponential form.

In the radical form, it is expressed as √55, whereas (55)^(1/2) in the exponential form. √55 ≈ 7.416198, 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 product of prime factors is the prime factorization of a number. Now let us look at how 55 is broken down into its prime factors.

Step 1: Finding the prime factors of 55

Breaking it down, we get 5 x 11: 5^1 x 11^1

Step 2: Now we found out the prime factors of 55. Since 55 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 55 using prime factorization is not straightforward.

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 55, we need to group it as 55.

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

Step 3: Add a decimal point and bring down two zeros to make the new dividend 600.

Step 4: Add the old divisor with the same number 7 + 7 to get 14, which will be our new divisor.

Step 5: The next step is finding 14n x n ≤ 600. Let us consider n as 4, then 144 x 4 = 576.

Step 6: Subtract 576 from 600; the difference is 24, and the quotient is 7.4.

Step 7: Since the dividend is less than the divisor, add another pair of zeros. The new dividend is 2400.

Step 8: Find the new divisor, which is 148, because 148 x 1 = 148.

Step 9: Subtracting 148 from 240 gives us 92.

Step 10: The quotient is now 7.41. Step 11: Continue doing these steps until we get the desired number of decimal places.

So the square root of √55 is approximately 7.4162.

The approximation method is another method for finding 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 55 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 55. The smallest perfect square less than 55 is 49, and the largest perfect square greater than 55 is 64. √55 falls between 7 and 8.

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 (55 - 49) / (64 - 49) = 0.4. 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 7 + 0.4 = 7.4, so the square root of 55 is approximately 7.4.

• 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.
   
• 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 the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• 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.