Square Root of 3360

The square root is the inverse of the square of a number. 3360 is not a perfect square. The square root of 3360 is expressed in both radical and exponential form. In radical form, it is expressed as √3360, whereas in exponential form it is expressed as (3360)^(1/2). √3360 ≈ 57.9332, 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 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 3360 is broken down into its prime factors.

Step 1: Finding the prime factors of 3360 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 5 x 7 x 7: 2^4 x 3 x 5 x 7^2

Step 2: Now we found the prime factors of 3360. The next step is to make pairs of those prime factors. Since 3360 is not a perfect square, the digits of the number can’t be grouped into pairs completely. Therefore, calculating √3360 using prime factorization directly is not feasible.

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, group the numbers from right to left. In the case of 3360, we need to group it as 33 and 60.

Step 2: Now find n whose square is less than or equal to 33. We can say n is '5' because 5 x 5 = 25, which is less than 33. The quotient is 5 after subtracting 25 from 33, the remainder is 8.

Step 3: Bring down 60, making the new dividend 860. Add the old divisor with the same number: 5 + 5 = 10, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 10n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 10n × n ≤ 860. Let us consider n as 8, now 108 x 8 = 864, which is greater than 860. Trying n as 7, we have 107 x 7 = 749.

Step 6: Subtract 749 from 860, the difference is 111, and the quotient is 57.

Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 11100.

Step 8: Find the new divisor that is 579 because 5799 x 9 = 52191.

Step 9: Subtracting 52191 from 11100 results in 889.

Step 10: Now the quotient is 57.9.

Step 11: Continue these steps until we get two numbers after the decimal point or until the remainder is zero. So the square root of √3360 is approximately 57.93.

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. Let us learn how to find the square root of 3360 using the approximation method.

Step 1: Find the closest perfect squares to √3360. The smallest perfect square less than 3360 is 3249 (57^2), and the largest perfect square more than 3360 is 3481 (59^2). √3360 falls between 57 and 59.

Step 2: Apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula: (3360 - 3249) ÷ (3481 - 3249) = 111 ÷ 232 ≈ 0.4784 Using the formula, we identified the decimal point of our square root. The next step is adding the initial integer to the decimal number: 57 + 0.4784 ≈ 57.93. So the square root of 3360 is approximately 57.93.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, and the inverse is the square root, √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction, meaning it cannot be written in the form of p/q, where p and q are integers and q ≠ 0.

• Principal square root: The positive square root of a number, which is used in most practical applications. It is the most commonly referenced square root.

• Perfect square: A number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

• Long division method: A method used to find the square root of a number that is not a perfect square, involving a series of steps to approximate the root.