Square Root of 359

The square root is the inverse of the square of the number. 359 is not a perfect square. The square root of 359 is expressed in both radical and exponential form. In the radical form, it is expressed as √359, whereas (359)^(1/2) in the exponential form. √359 ≈ 18.947, 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 359 is broken down into its prime factors.

Step 1: Finding the prime factors of 359 359 is a prime number, so it cannot be broken down further into other prime factors.

Step 2: Since 359 is not a perfect square, calculating the square root using prime factorization is not applicable.

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 359, we need to group it as 59 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.

Step 3: Subtract 1 from 3, and the remainder is 2. Bring down 59, making the new dividend 259.

Step 4: The new divisor will be 2 times the current quotient, which is 2.

Step 5: Find n, such that 2n x n ≤ 259. Let us consider n as 9, now 29 x 9 = 261. Since 261 is greater than 259, consider n as 8.

Step 6: Subtract 224 from 259, and the difference is 35.

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 zeros to the dividend. Now the new dividend is 3500.

Step 8: Find the new divisor starting with 28, making it 288, and find n such that 288n x n ≤ 3500.

Step 9: Continue the process until the required precision is achieved.

The square root of 359 is approximately 18.947.

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 359 using the approximation method.

Step 1: Find the closest perfect squares around 359. The smallest perfect square less than 359 is 324, and the largest perfect square greater than 359 is 361. Therefore, √359 falls between 18 and 19.

Step 2: Apply the formula:

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

(359 - 324) / (361 - 324) = 35 / 37 ≈ 0.946.

Step 3: Add the decimal to the smaller integer: 18 + 0.946 = 18.946.

Therefore, the square root of 359 is approximately 18.947.

• 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, which 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.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
   
• Approximation: The process of finding a value that is close enough to the correct answer, usually within an acceptable error range.
   
• Long division method: A step-by-step method used to divide larger numbers and to find roots of numbers, particularly when the number is not a perfect square.