Square Root of 2064

The square root is the inverse of the square of a number. 2064 is not a perfect square. The square root of 2064 is expressed in both radical and exponential form. In the radical form, it is expressed as √2064, whereas (2064)^(1/2) in the exponential form. √2064 ≈ 45.433, 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 2064 is broken down into its prime factors.

Step 1: Finding the prime factors of 2064

Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 7 x 7: 2^3 x 3^2 x 7^2

Step 2: Now we found out the prime factors of 2064. The second step is to make pairs of those prime factors. Since 2064 is not a perfect square, therefore the digits of the number can’t be grouped in complete pairs. Therefore, calculating √2064 using prime factorization gives us an approximate value since it contains unpaired factors.

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 2064, we need to group it as 64 and 20.

Step 2: Now we need to find n whose square is less than or equal to 20. We can say n as ‘4’ because 4 x 4 is equal to 16, which is less than 20. Now the quotient is 4, after subtracting 16 from 20 the remainder is 4.

Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor with the same number: 4 + 4 equals 8, which will be our new divisor.

Step 4: The new divisor will be 8n. We need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 464. Let us consider n as 5, now 85 x 5 = 425.

Step 6: Subtract 425 from 464, the difference is 39, and the quotient is 45.

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 3900.

Step 8: Now we need to find the new divisor by considering a number for n that satisfies 900n x n ≤ 3900.

Step 9: Through approximation, we find n, and continue the process to get the square root up to the desired decimal places.

So the square root of √2064 is approximately 45.433.

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 2064 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2064. The smallest perfect square less than 2064 is 2025 and the largest perfect square greater than 2064 is 2116. √2064 falls somewhere between 45 and 46.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (2064 - 2025) / (2116 - 2025) = 39 / 91 ≈ 0.428

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 45 + 0.428 = 45.428, so the square root of 2064 is approximately 45.428.

• 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 always the positive square root that is more prominent due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is breaking a number down into the set of prime numbers which multiply together to give the original number.
   
• Approximation: Approximation is a method of finding a value that is close enough to the correct answer, usually with some thought or calculation.