Square Root of 213

The square root is the inverse of the square of the number. 213 is not a perfect square. The square root of 213 is expressed in both radical and exponential form. In the radical form, it is expressed as √213, whereas (213)^(1/2) in the exponential form. √213 ≈ 14.5945, 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, for non-perfect square numbers, 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 213 is broken down into its prime factors:

Step 1: Finding the prime factors of 213 Breaking it down, we get 3 x 71: 3^1 x 71^1

Step 2: Now we have found the prime factors of 213. Since 213 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √213 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 213, we group it as 13 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 less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 13, which is the new dividend. Add the old divisor with the same number: 1 + 1 = 2, which will be our new divisor.

Step 4: The new divisor will be 2n. We need to find the value of n.

Step 5: Find 2n × n ≤ 113. Let us consider n as 4, now 24 x 4 = 96.

Step 6: Subtract 96 from 113; the difference is 17, and the quotient is 14.

Step 7: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1700.

Step 8: Find the new divisor, let’s say it is 145, because 145 x 5 = 725.

Step 9: Subtracting 725 from 1700 gives the result 975.

Step 10: Now the quotient is 14.5

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

So the square root of √213 is approximately 14.59.

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 213 using the approximation method.

Step 1: Find the closest perfect squares to √213. The smallest perfect square less than 213 is 196, and the largest perfect square greater than 213 is 225. √213 falls between 14 and 15.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (213 - 196) ÷ (225 - 196) = 17 ÷ 29 ≈ 0.5862. Adding the initial integer value to the decimal, 14 + 0.59 = 14.59, so the square root of 213 is approximately 14.59.

• 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, which 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; the positive square root is often used in practical applications. This is known as the principal square root.

• Integer: An integer is a whole number that can be positive, negative, or zero. Examples include -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.

• Approximation: Approximation is a method used to find a value that is close to the actual value. For example, the square root of 213 is approximately 14.59.