Square Root of 334

The square root is the inverse of the square of a number. 334 is not a perfect square. The square root of 334 is expressed in both radical and exponential forms. In the radical form, it is expressed as √334, whereas in exponential form it is (334)^(1/2). √334 ≈ 18.276, 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, the prime factorization method is typically not used; instead, 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 334 is broken down into its prime factors:

Step 1: Finding the prime factors of 334 Breaking it down, we get 2 x 167: 2^1 x 167^1

Step 2: We found the prime factors of 334. The second step is to make pairs of those prime factors. Since 334 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating 334 using prime factorization alone is not possible.

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 334, we need to group it as 34 and 3.

Step 2: Now we need to find n whose square is 3. We can say n is ‘1’ because 1 x 1 is less 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 34, 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, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 234. Let us consider n as 9, now 2 x 9 x 9 = 162.

Step 6: Subtracting 162 from 234 gives the difference of 72, 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 7200.

Step 8: Now we need to find a new divisor.

Step 9: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √334 ≈ 18.276

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 334 using the approximation method.

Step 1: Now we have to find the closest perfect square of √334.

The smallest perfect square less than 334 is 324, and the largest perfect square greater than 334 is 361. √334 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula, (334 - 324) / (361 - 324) = 10 / 37 ≈ 0.27 Adding this value to the lower perfect square root: 18 + 0.27 = 18.27

Therefore, the approximate square root of 334 is 18.27

• 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 expressed as a fraction 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, but the principal square root is the non-negative square root that is typically more useful in real-world applications.

• Prime factorization: This is the process of expressing a number as the product of its prime factors.

• Approximation method: A method used to find the approximate value of the square root of a non-perfect square by estimating between two perfect squares.