Square Root of 221

The square root is the inverse of the square of the number. 221 is not a perfect square. The square root of 221 is expressed in both radical and exponential form. In the radical form, it is expressed as √221, whereas (221)^(1/2) in the exponential form. √221 ≈ 14.86607, 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 221 is broken down into its prime factors:

Step 1: Finding the prime factors of 221 Breaking it down, we get 13 x 17.

Step 2: Now we found out the prime factors of 221. The second step is to make pairs of those prime factors. Since 221 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 221 using prime factorization is impossible.

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 221, we need to group it as 21 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n is ‘1’ because 1 x 1 is less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 121. Let us consider n as 5, now 25 x 5 = 125, which is too large. Trying n as 4, we have 24 x 4 = 96, which is less than 121.

Step 6: Subtract 96 from 121, and the difference is 25. The quotient is now 14.

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 2500.

Step 8: Now we need to find the new divisor, which is 298, because 298 x 8 = 2384.

Step 9: Subtracting 2384 from 2500 we get the result 116. Step 10: Now the quotient is 14.8.

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 √221 is approximately 14.86.

The 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 221 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √221. The smallest perfect square less than 221 is 196 (14^2) and the largest perfect square greater than 221 is 225 (15^2). √221 falls somewhere between 14 and 15.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (221 - 196) ÷ (225 - 196) = 25 ÷ 29 ≈ 0.86 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 14 + 0.86 = 14.86, so the square root of 221 is approximately 14.86.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4^2 = 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.

• 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 expressing a number as the product of its prime numbers. Example: 221 = 13 x 17.

• 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.