Square Root of 211

The square root is the inverse of the square of the number. 211 is not a perfect square. The square root of 211 is expressed in both radical and exponential forms. In the radical form, it is expressed as √211, whereas (211)^(1/2) in the exponential form. √211 ≈ 14.525839, 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 211 is broken down into its prime factors.

Step 1: Finding the prime factors of 211 211 is a prime number, so it cannot be broken down into smaller prime factors.

Since 211 is not a perfect square, calculating √211 using prime factorization is not feasible.

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 211, we need to group it as 11 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 × 1 is less than or equal to 2. Now the quotient is 1; after subtracting 1 × 1 from 2, the remainder is 1.

Step 3: Now let us bring down 11, 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 next step is finding 2n × n ≤ 111. Let us consider n as 5, now 25 × 5 = 125.

Step 5: Since 125 is greater than 111, try n as 4, now 24 × 4 = 96.

Step 6: Subtract 96 from 111; the difference is 15, and the quotient becomes 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 1500.

Step 8: Now we need to find the new divisor that is 289 because 289 × 5 = 1445.

Step 9: Subtracting 1445 from 1500, we get the result 55.

Step 10: Now the quotient is 14.5.

Step 11: Continue doing these steps until we get the desired precision.

So the square root of √211 is approximately 14.525839.

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 211 using the approximation method.

Step 1: Now we have to find the closest perfect square of √211. The smallest perfect square less than 211 is 196 (14^2), and the largest perfect square greater than 211 is 225 (15^2). √211 falls somewhere between 14 and 15.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (211 - 196) / (225 - 196) = 15 / 29 ≈ 0.517. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the integer, which is 14 + 0.517 ≈ 14.517.

Therefore, the square root of 211 is approximately 14.525839.

• 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.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Long division method: A method used to find the square root of non-perfect squares by a systematic division process.

• Approximation method: A method used to estimate the square root of a number by identifying its closest perfect squares and calculating the decimal value.