Square Root of 318

The square root is the inverse of the square of the number. 318 is not a perfect square. The square root of 318 is expressed in both radical and exponential form. In the radical form, it is expressed as √318, whereas (318)^(1/2) in the exponential form. √318 ≈ 17.8326, 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 318 is broken down into its prime factors.

Step 1: Finding the prime factors of 318 Breaking it down, we get 2 x 3 x 53: 2¹ x 3¹ x 53¹

Step 2: Now we found out the prime factors of 318. The second step is to make pairs of those prime factors. Since 318 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √318 using prime factorization directly 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 318, we need to group it as 18 and 3.

Step 2: Now we need to find n whose square is ≤ 3. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 3. Now the quotient is 1 after subtracting 3 - 1 the remainder is 2.

Step 3: Now let us bring down 18 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 × n ≤ 218. Let us consider n as 8, now 28 x 8 = 224; we need a value less than 218, so n is 7.

Step 6: Subtract 218 from 217 the difference is 1, and the quotient is 17.

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 100.

Step 8: Now we need to find the new divisor that is 347 because 347 x 2 = 694.

Step 9: Subtracting 694 from 1000 we get the result 306.

Step 10: Now the quotient is 17.83.

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 √318 is approximately 17.83.

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 318 using the approximation method.

Step 1: Now we have to find the closest perfect square of √318.

The smallest perfect square less than 318 is 289 (17²) and the largest perfect square greater than 318 is 324 (18²). √318 falls somewhere between 17 and 18.

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: (318 - 289) ÷ (324 - 289) = 29 ÷ 35 ≈ 0.8286.

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 17 + 0.8286 ≈ 17.83, so the square root of 318 is approximately 17.83.

• 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4².

• Approximation method: This is a method used to estimate the square root of non-perfect squares by comparing them to nearby perfect squares.

• Long division method: A method used to calculate more accurate values of square roots for non-perfect squares through a systematic division process.