Square Root of 306

The square root is the inverse of the square of the number. 306 is not a perfect square. The square root of 306 is expressed in both radical and exponential form. In the radical form, it is expressed as √306, whereas (306)^(1/2) in the exponential form. √306 ≈ 17.49285, 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 306 is broken down into its prime factors.

Step 1: Finding the prime factors of 306

Breaking it down, we get 2 x 3 x 3 x 17: 2^1 x 3^2 x 17^1

Step 2: Now we found out the prime factors of 306. The second step is to make pairs of those prime factors. Since 306 is not a perfect square, therefore the digits of the number can’t be grouped in pair. Therefore, calculating 306 using prime factorization is not straightforward.

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 306, we need to group it as 06 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is '1' because 1 x 1 is lesser than or equal to 3. Now the quotient is 1. After subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 06 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 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 ≤ 206. Let us consider n as 8. Now 28 x 8 = 224, which is too large. Trying n as 7 works since 27 x 7 = 189.

Step 6: Subtract 189 from 206. The difference is 17, 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 1700.

Step 8: Now we need to find the new divisor. If the new divisor is 354, then 354 x 4 = 1416, which fits into 1700.

Step 9: Subtracting 1416 from 1700, we get the result 284.

Step 10: Now the quotient is 17.4.

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 √306 is 17.49.

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 306 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √306. The closest perfect square less than 306 is 289, and the closest perfect square greater than 306 is 324. √306 falls somewhere between 17 and 18.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (306 - 289) ÷ (324 - 289) = 17 ÷ 35 ≈ 0.486. 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 + 0.49 = 17.49. 17 + 0.49 = 17.49, so the square root of 306 is approximately 17.49.

• 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 a principal square root.

• Prime factorization: Expressing a number as a product of its prime factors. For example, 306 = 2 x 3 x 3 x 17.

• Approximation method: A technique to find an approximate value for the square root of a number by using nearby perfect squares.