Square Root of 493

The square root is the inverse of the square of a number. 493 is not a perfect square. The square root of 493 is expressed in both radical and exponential form.

In the radical form, it is expressed as √493, whereas (493)^(1/2) in the exponential form. √493 ≈ 22.205, 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 prime factorization method is not used; instead, long-division and approximation methods are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's look at how 493 is broken down into its prime factors.

Step 1: Finding the prime factors of 493

Breaking it down, we get 17 x 29, as 493 = 17 x 29.

Step 2: Since 493 is not a perfect square, its prime factors cannot be grouped into pairs.

Therefore, calculating 493 using prime factorization alone 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, we need to group the numbers from right to left. In the case of 493, we consider 93 and 4.

Step 2: Now we need to find n whose square is less than or equal to 4. We can say n is 2 because 2 x 2 = 4. Now the quotient is 2, and after subtracting 4, the remainder is 0.

Step 3: Bring down 93, which is the new dividend. Add the old divisor with the same number (2), and we get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the previous divisor and quotient. We need to find n such that 4n x n ≤ 93.

Step 5: Consider n as 2, then 42 x 2 = 84.

Step 6: Subtract 84 from 93, the difference is 9, and the quotient is 22.

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 900.

Step 8: Find the new divisor, which is 445 because 445 x 2 = 890.

Step 9: Subtracting 890 from 900, we get the result 10.

Step 10: Now the quotient is 22.2.

Step 11: Continue these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √493 is approximately 22.205.

The approximation method is another way to find square roots, and it is an easy method to find the square root of a given number. Let us learn how to find the square root of 493 using the approximation method.

Step 1: Find the closest perfect squares to √493. The smallest perfect square less than 493 is 484, and the largest perfect square greater than 493 is 529. √493 falls between 22 and 23.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (493 - 484) / (529 - 484) ≈ 0.205. Adding this decimal to 22 (the square root of 484), we get 22 + 0.205 = 22.205, so the square root of 493 is approximately 22.205.

• 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.
   
• Radical: A radical is the symbol √ used to denote the square root or nth root of a number.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Long division method: The long division method is a step-by-step process used for finding the square root of numbers that are not perfect squares.