Square Root of 5500

The square root is the inverse of the square of the number. 5500 is not a perfect square. The square root of 5500 is expressed in both radical and exponential form. In radical form, it is expressed as √5500, whereas (5500)^(1/2) in exponential form. √5500 = 74.16198, 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 the 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 5500 is broken down into its prime factors.

Step 1: Finding the prime factors of 5500 Breaking it down, we get 2 x 2 x 5 x 5 x 11 x 5: 2^2 x 5^3 x 11

Step 2: Now we found out the prime factors of 5500. The second step is to make pairs of those prime factors. Since 5500 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √5500 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 5500, we need to group it as 55 and 00.

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

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number, 7 + 7, we get 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 14n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 14n x n ≤ 600. Let n be 4, now 144 x 4 = 576.

Step 6: Subtract 576 from 600, the difference is 24, and the quotient is 74.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2400.

Step 8: Now we need to find the new divisor. Let n be 1, because 1481 x 1 = 1481.

Step 9: Subtracting 1481 from 2400, we get the result 919.

Step 10: Now the quotient is 74.1

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √5500 is approximately 74.16.

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 5500 using the approximation method.

Step 1: Now we have to find the closest perfect square of √5500.

The smallest perfect square less than 5500 is 5476 and the largest perfect square greater than 5500 is 5625. √5500 falls somewhere between 74 and 75.

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 (5500 - 5476) ÷ (5625 - 5476) = 24 ÷ 149 ≈ 0.161

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 74 + 0.161 ≈ 74.161, so the square root of 5500 is approximately 74.161.

• Square root: A square root is the inverse of a square. For 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 the principal square root.
   
• Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 5500 is 2^2 x 5^3 x 11.
   
• Long division method: The long division method is a technique for determining the square root of a number by dividing and finding the average of divisors and dividends.