Square Root of 208

The square root is the inverse of the square of the number. 208 is not a perfect square. The square root of 208 is expressed in both radical and exponential form. In the radical form, it is expressed as √208, whereas 208^(1/2) in the exponential form. √208 ≈ 14.422, 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 208 is broken down into its prime factors:

Step 1: Finding the prime factors of 208 Breaking it down, we get 2 x 2 x 2 x 2 x 13: 2^4 x 13

Step 2: Now we found out the prime factors of 208. The second step is to make pairs of those prime factors. Since 208 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating 208 using prime factorization only gives an approximation.

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 208, we need to group it as 08 and 2.

Step 2: Now we need to find n whose square is 1. 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 x 1 from 2, the remainder is 1.

Step 3: Now let us bring down 08, which is the new dividend. Add the old divisor with the same number: 1 + 1, which gives 2, 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 ≤ 108. Let us consider n as 4, now 24 x 4 = 96.

Step 6: Subtract 108 from 96; the difference is 12, 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 1200.

Step 8: Now we need to find the new divisor: 289, because 289 x 4 = 1156.

Step 9: Subtracting 1156 from 1200, we get the result 44.

Step 10: Now the quotient is 14.4

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

So the square root of √208 is approximately 14.42.

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 208 using the approximation method.

Step 1: Now we have to find the closest perfect square of √208. The smallest perfect square near 208 is 196, and the largest perfect square near 208 is 225. √208 falls somewhere between 14 and 15.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Going by the formula (208 - 196) / (225 - 196) = 0.41 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: 14 + 0.41 = 14.41, so the square root of 208 is approximately 14.41

• 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.

• 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: It is the expression of a number as the product of its prime factors. Example: The prime factorization of 208 is 2^4 x 13.

• 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.