Square Root of 2564

The square root is the inverse of the square of the number. 2564 is not a perfect square. The square root of 2564 is expressed in both radical and exponential form. In the radical form, it is expressed as √2564, whereas (2564)^(1/2) in the exponential form. √2564 ≈ 50.63596, 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, methods like 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 2564 is broken down into its prime factors.

Step 1: Finding the prime factors of 2564 Breaking it down, we get 2 x 2 x 641: 2^2 x 641

Step 2: Now we have found the prime factors of 2564. The next step is to make pairs of those prime factors. Since 2564 is not a perfect square, the digits of the number can’t be grouped into pairs completely.

Therefore, calculating √2564 using prime factorization is not straightforward.

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 2564, we group it as 25 and 64.

Step 2: Now we need to find n whose square is ≤ 25. We say n is ‘5’ because 5 x 5 = 25. Now the quotient is 5, and after subtracting 25 - 25, the remainder is 0.

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

Step 4: Now we get 10n as the new divisor, and we need to find the value of n such that 10n x n ≤ 64. We consider n as 6, now 106 x 6 = 636.

Step 5: Since 64 is less than 636, 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 6400.

Step 6: Now we need to find the new divisor. Continuing the division process will yield the quotient incrementally.

Step 7: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √2564 is approximately 50.64.

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. Now let us learn how to find the square root of 2564 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2564. The smallest perfect square less than 2564 is 2500, and the largest perfect square greater than 2564 is 2601. √2564 falls somewhere between 50 and 51.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (2564 - 2500) ÷ (2601 - 2500) ≈ 0.64 Adding the initial integer approximation to the decimal number, 50 + 0.64 = 50.64'

So the square root of 2564 is approximately 50.64.

• Square root: A square root is the inverse of a square. Example: 6^2 = 36, and the inverse of the square is the square root, that is, √36 = 6.
   
• 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, the positive square root is often used in practical applications. This is known as the principal square root.
   
• Long division method: A method used to find the square root of non-perfect square numbers by dividing and approximating.
   
• Radical form: The expression of a square root using the radical (√) symbol, such as √2564.