Square Root of 214

The square root is the inverse of the square of a number. 214 is not a perfect square. The square root of 214 is expressed in both radical and exponential form. In the radical form, it is expressed as √214, whereas \(214^{1/2}\) in the exponential form. √214 ≈ 14.62874, 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 214 is broken down into its prime factors:

Step 1: Finding the prime factors of 214 Breaking it down, we get 2 x 107: (2^1) x (107^1).

Step 2: Now we found out the prime factors of 214. The second step is to make pairs of those prime factors. Since 214 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 214 using prime factorization is not feasible for an exact square root.

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 214, we need to group it as 14 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n as ‘1’ because 1 x 1 is lesser 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 14, which is the new dividend. Add the old divisor with the same number 1 + 1, we 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, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 114. Let us consider n as 5, now 25 x 5 = 125, which is too large. Trying n = 4, we find 24 x 4 = 96.

Step 6: Subtract 96 from 114; the difference is 18, and the quotient is 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 1800.

Step 8: Now we need to find the new divisor that is 289 because 289 x 6 = 1734.

Step 9: Subtracting 1734 from 1800, we get the result 66.

Step 10: Now the quotient is 14.6.

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

So the square root of √214 is approximately 14.63.

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 214 using the approximation method:

Step 1: Now we have to find the closest perfect squares of √214. The smallest perfect square less than 214 is 196 (14²) and the largest perfect square greater than 214 is 225 (15²). √214 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). Applying the formula: (214 - 196) / (225 - 196) = 18 / 29 ≈ 0.62. 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 + 0.62 = 14.62. So, the square root of 214 is approximately 14.62.

• 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 usually the positive square root that is used in most real-world applications, which is why it is known as the principal square root.

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

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs from right to left and finding the root step by step.