Square Root of 364

The square root is the inverse of the square of a number. 364 is not a perfect square. The square root of 364 is expressed in both radical and exponential form. In radical form, it is expressed as √364, whereas (364)^(1/2) in exponential form. √364 ≈ 19.07878, 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 the 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 364 is broken down into its prime factors:

Step 1: Finding the prime factors of 364. Breaking it down, we get 2 × 2 × 7 × 13: 2^2 × 7^1 × 13^1.

Step 2: Now we have found out the prime factors of 364. The second step is to make pairs of those prime factors. Since 364 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 364 using prime factorization is not feasible for finding its square root directly.

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 364, we need to group it as 64 and 3.

Step 2: Now we need to find n whose square is 3. We can say n is ‘1’ because 1 × 1 is lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 64, 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 ≤ 264. Let us consider n as 9, now 29 × 9 = 261.

Step 6: Subtract 261 from 264; the difference is 3, and the quotient is 19.

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 300.

Step 8: Now we need to find the new divisor, which is 381, because 381 × 1 = 381.

Step 9: Subtracting 381 from 400 results in 19.

Step 10: Now the quotient is 19.0.

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

So the square root of √364 is approximately 19.08.

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Now let us learn how to find the square root of 364 using the approximation method.

Step 1: Now we have to find the closest perfect square to √364. The smallest perfect square less than 364 is 361, and the largest perfect square greater than 364 is 400. √364 falls between 19 and 20.

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 (364 - 361) / (400 - 361) = 3 / 39 ≈ 0.0769.

Using the formula, we identified the decimal point of our square root.

The next step is adding the initial value to the decimal number, which is 19 + 0.0769 ≈ 19.08.

• Square root: A square root is the inverse of a square. For 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: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Long division method: This is a method used to find the square root of non-perfect squares by dividing the number into smaller, more manageable parts.