Square Root of 577

The square root is the inverse of the square of the number. 577 is not a perfect square. The square root of 577 is expressed in both radical and exponential form. In radical form, it is expressed as √577, whereas in exponential form it is expressed as (577)^(1/2). √577 ≈ 24.0208, 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, the long division method and the 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 577 is broken down into its prime factors:

Step 1: Finding the prime factors of 577 577 is a prime number, meaning it cannot be broken down further into smaller prime factors.

Therefore, prime factorization is not applicable here.

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 digits of 577 from right to left as 77 and 5.

Step 2: Now we need to find n whose square is closest to 5. We can say n as ‘2’ because 2 × 2 = 4 is less than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Bring down 77, making the new dividend 177. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: Find n such that 4n × n ≤ 177. Let us consider n as 4. Now, 44 × 4 = 176.

Step 5: Subtract 176 from 177, the difference is 1, and the quotient is 24.

Step 6: 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, making it 100.

Step 7: Find the new divisor 480 because 480 × 0 = 0.

Step 8: Subtract 0 from 100, and continue the process until you achieve the desired precision. The square root of √577 is approximately 24.0208.

The approximation method is another way to find 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 577 using the approximation method.

Step 1: Find the closest perfect squares around 577. The smallest perfect square before 577 is 576, and the next perfect square is 625. √577 falls between 24 and 25.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (577 - 576) / (625 - 576) = 0.02

Step 3: Add the value obtained to the smaller square root: 24 + 0.02 = 24.02, so the square root of 577 is approximately 24.02.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction, i.e., in the form of p/q, where q is not equal to zero and p and q are integers.

• Prime number: A number greater than 1 that has no positive divisors other than 1 and itself. Example: 577.

• Long division method: A technique used to find the approximate square root of a number, especially for non-perfect squares.

• Decimal approximation: The process of finding an approximate decimal value for a square root that is not a perfect square.