Square Root of 558

The square root is the inverse of the square of the number. 558 is not a perfect square. The square root of 558 is expressed in both radical and exponential form. In the radical form, it is expressed as √558, whereas (558)^(1/2) in the exponential form. √558 ≈ 23.622, 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:

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 558 is broken down into its prime factors.

Step 1: Finding the prime factors of 558 Breaking it down, we get 2 × 3 × 3 × 31: 2¹ × 3² × 31¹

Step 2: Now we have found the prime factors of 558. The second step is to make pairs of those prime factors.

Since 558 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating the square root of 558 using prime factorization is impossible.

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 558, we need to group it as 58 and 5.

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

Step 3: Now let us bring down 58, 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 next step is finding 4n × n ≤ 158. Let us consider n as 3, now 43 × 3 = 129.

Step 5: Subtract 129 from 158. The difference is 29, and the quotient is 23.

Step 6: 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 2900.

Step 7: Now we need to find the new divisor that is 6 because 466 × 6 = 2796.

Step 8: Subtracting 2796 from 2900, we get the result 104.

Step 9: Now the quotient is 23.6.

Step 10: 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 √558 is approximately 23.62.

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 558 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √558. The smallest perfect square less than 558 is 529, and the largest perfect square greater than 558 is 576. √558 falls somewhere between 23 and 24.

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 (558 - 529) ÷ (576 - 529) = 29 ÷ 47 ≈ 0.617 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 + 0.617 ≈ 23.617.

Therefore, the square root of 558 is approximately 23.617.

• 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, 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 has more prominence due to its uses in the real world. That is why it is also known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors, which are the building blocks of all numbers.

• 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.