Square Root of 341

The square root is the inverse of the square of the number. 341 is not a perfect square. The square root of 341 is expressed in both radical and exponential forms. In the radical form, it is expressed as √341, whereas (341)^(1/2) in the exponential form. √341 ≈ 18.46619, 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 long-division and approximation methods 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 341 is broken down into its prime factors.

Step 1: Finding the prime factors of 341 Breaking it down, we get 341 = 11 x 31.

Step 2: Now we found out the prime factors of 341. Since 341 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √341 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 group the numbers from right to left. In the case of 341, we group it as 41 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n as ‘1’ because 1 x 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: Bring down 41, 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 2, and we need to find 2n x n ≤ 241. Let’s consider n as 8, then 28 x 8 = 224.

Step 5: Subtract 224 from 241, the difference is 17, and the quotient is 18.

Step 6: 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 7: Now we need to find the new divisor. The new divisor is 369 because 369 x 4 = 1476.

Step 8: Subtract 1476 from 1700, and the remainder is 224.

Step 9: Continue this method until the desired precision is achieved.

So the square root of √341 ≈ 18.47

The approximation method is another method for finding the 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 341 using the approximation method.

Step 1: Find the closest perfect squares to √341.

The smallest perfect square less than 341 is 324, and the largest perfect square greater than 341 is 361. √341 falls between √324 = 18 and √361 = 19.

Step 2: Now apply the formula [(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)].

Using the formula, (341 - 324) / (361 - 324) ≈ 0.459. Adding this to the lower bound, 18 + 0.459 ≈ 18.459.

So the square root of 341 is approximately 18.47.

• 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, the positive square root is more commonly used due to its applications in the real world. It is known as the principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 9 is a perfect square because it is 3^2.

• Long division method: A method used to find the square root of non-perfect squares by performing division in a systematic way until the desired precision is achieved.