Square Root of 3400

The square root is the inverse of the square of the number. 3400 is not a perfect square. The square root of 3400 is expressed in both radical and exponential form. In the radical form, it is expressed as √3400, whereas (3400)^(1/2) is its exponential form. √3400 ≈ 58.3095, 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 3400 is broken down into its prime factors.

Step 1: Finding the prime factors of 3400 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 17: 2^3 x 5^2 x 17

Step 2: Now we found the prime factors of 3400. The second step is to make pairs of those prime factors. Since 3400 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 3400 using prime factorization is limited.

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 3400, we need to group it as 00 and 34.

Step 2: Now, we need to find n whose square is less than or equal to 34. We can say n is '5' because 5 x 5 = 25, which is less than 34. The quotient is 5, and after subtracting 25 from 34, the remainder is 9.

Step 3: Bring down the next pair of digits, which is 00, to make the new dividend 900.

Step 4: Add the old divisor to itself, 5 + 5, to get 10, which becomes our new divisor.

Step 5: Find a digit 'd' such that (10d) x d is less than or equal to 900. In this case, d is 8, since 108 x 8 = 864.

Step 6: Subtract 864 from 900 to get 36 as the remainder, and the quotient is now 58.

Step 7: Since the dividend is less than the divisor, we add a decimal point and continue the division. Add two zeroes to the dividend, making it 3600.

Step 8: The new divisor becomes 580. Finding the suitable digit gives us the quotient 58.3. Continue this process to find more decimal places as needed. So the square root of √3400 is approximately 58.3095.

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 3400 using the approximation method.

Step 1: Find the closest perfect squares around 3400. The closest perfect squares are 3364 (58^2) and 3481 (59^2). √3400 falls between 58 and 59.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (3400 - 3364) / (3481 - 3364) ≈ 0.365 The approximation step gives us the decimal value. Adding this to the base integer, 58 + 0.365 = 58.365, so the square root of 3400 is approximately 58.365.

• Square root: A square root is the number that, when multiplied by itself, gives the original number. Example: The square root of 25 is 5 because 5 x 5 = 25.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction. Its decimal form is non-repeating and non-terminating.

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

• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime numbers. For example, the prime factorization of 18 is 2 x 3 x 3.

• Long division method: The long division method is a technique to find the square root of a number by dividing and averaging iteratively.