Square Root of 2664

The square root is the inverse of the square of the number. 2664 is not a perfect square. The square root of 2664 is expressed in both radical and exponential form. In the radical form, it is expressed as √2664, whereas (2664)^(1/2) in the exponential form. √2664 ≈ 51.627, 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 2664 is broken down into its prime factors.

Step 1: Finding the prime factors of 2664 Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 37: 2^3 x 3^2 x 37

Step 2: Now we found out the prime factors of 2664. The second step is to make pairs of those prime factors. Since 2664 is not a perfect square, the digits of the number can’t be grouped in a perfect pair.

Therefore, calculating 2664 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, we need to group the numbers from right to left. In the case of 2664, we need to group it as 64 and 26.

Step 2: Now we need to find n whose square is closest to 26. We can use n = 5 because 5 x 5 = 25 is less than 26. Now the quotient is 5, and after subtracting 25 from 26, the remainder is 1.

Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor with the same number 5 + 5, giving us 10, which will be our new divisor.

Step 4: The new divisor will be 10, and we need to find a digit to replace the blank in 10_ that, when multiplied by itself, results in a product less than or equal to 164.

Step 5: The next step is finding 10n × n ≤ 164. Let us consider n as 1, now 101 x 1 = 101.

Step 6: Subtract 101 from 164, the difference is 63, and the quotient is 51.

Step 7: Since the remainder is less than the divisor, we add a decimal point and bring down two zeros to make the dividend 6300.

Step 8: Now, the new divisor is 102, and we continue the process as with integers, finding the next digit of the quotient. Continue doing these steps until we get the desired number of decimal places.

So the approximate square root of √2664 is 51.627.

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 2664 using the approximation method.

Step 1: Now we have to find the closest perfect square to √2664. The closest perfect squares around 2664 are 2601 (51^2) and 2704 (52^2). √2664 falls somewhere between 51 and 52.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Going by the formula (2664 - 2601) ÷ (2704 - 2601) = 63 ÷ 103 ≈ 0.611 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 51 + 0.611 = 51.611.

Thus, the approximate square root of 2664 is 51.611.

• 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, it is the positive square root that is most commonly used, known as the principal square root.
   
• Prime factorization: A method of expressing a number as the product of its prime factors.
   
• Decimal approximation: An estimate of an irrational number's value using decimals, often used when an exact expression is complex or impossible to obtain in simple form.