Square Root of 557

The square root is the inverse of the square of the number. 557 is not a perfect square. The square root of 557 is expressed in both radical and exponential form. In the radical form, it is expressed as √557, whereas (557)^(1/2) in the exponential form. √557 ≈ 23.598, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 557 is broken down into its prime factors.

Step 1: Finding the prime factors of 557

557 is a prime number itself, so it cannot be further broken down into smaller prime numbers.

Step 2: Since 557 is not a perfect square, calculating its square root using prime factorization 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 557, we need to group it as 57 and 5.

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

Step 3: Now let us bring down 57 which is the new dividend. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of 40 and n. Now we get 4n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 157. Let us consider n as 3, now 43 × 3 = 129.

Step 6: Subtract 129 from 157, the difference is 28, and the quotient is 23.

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 2800.

Step 8: Now we need to find the new divisor that is 236 because 2366 × 6 = 14196.

Step 9: Subtracting 14196 from 28000 we get the result 13804.

Step 10: Now the quotient is 23.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √557 ≈ 23.60.

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 557 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √557. The smallest perfect square less than 557 is 529, and the largest perfect square greater than 557 is 576. √557 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (557 - 529) ÷ (576 - 529) = 28 ÷ 47 ≈ 0.596.

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 23 + 0.596 ≈ 23.596, so the square root of 557 is approximately 23.596.

• Square root: A square root is the inverse of a square. Example: 4² = 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.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Approximation: The process of finding a value that is close enough to the correct answer, usually within a specified limit.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.