Square Root of 369

The square root is the inverse of the square of the number. 369 is not a perfect square. The square root of 369 is expressed in both radical and exponential form. In the radical form, it is expressed as √369, whereas (369)^(1/2) in the exponential form. √369 ≈ 19.209, 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, methods such as 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 369 is broken down into its prime factors.

Step 1: Finding the prime factors of 369 Breaking it down, we get 3 x 3 x 41: 3² x 41

Step 2: Now that we have found the prime factors of 369, the second step is to make pairs of those prime factors. Since 369 is not a perfect square, complete pairing of digits is not possible.

Therefore, calculating 369 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 369, we group it as 69 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 ≤ 3. Now, the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 69 to make 269, 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 ≤ 269. Let us consider n as 9, now 29 x 9 = 261.

Step 6: Subtract 261 from 269, the difference is 8, and the quotient becomes 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 800.

Step 8: Now we need to find the new divisor. Consider n as 3, giving us 193 x 3 = 579.

Step 9: Subtracting 579 from 800 gives us the result 221.

Step 10: Now the quotient is 19.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point, or until the remainder is zero.

So the square root of √369 ≈ 19.21.

The approximation method is another method for finding square roots, and 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 369 using the approximation method.

Step 1: Find the closest perfect squares to √369.

The closest perfect square less than 369 is 361, and the closest perfect square greater than 369 is 400.

√369 falls somewhere between 19 and 20.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (369 - 361) / (400 - 361) = 8 / 39 ≈ 0.205.

By adding the value 19 (the square root of 361) to 0.205, we get approximately 19.205.

So the square root of 369 is approximately 19.21.

• 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.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is often used due to its applications in the real world. Hence, it is known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Long division method: A step-by-step method used to find the square root of non-perfect square numbers by dividing the number into groups of digits.