Square Root of 496

The square root is the inverse of the square of a number. 496 is not a perfect square. The square root of 496 is expressed in both radical and exponential form. In radical form, it is expressed as √496, whereas in exponential form it is expressed as (496)^(1/2). √496 ≈ 22.27106, 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.

For perfect square numbers, the prime factorization method can be used. However, for non-perfect square numbers, methods such as 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 prime factorization of a number is the product of its prime factors. Now let us look at how 496 is broken down into its prime factors.

Step 1: Finding the prime factors of 496 Breaking it down, we get 2 x 2 x 2 x 2 x 31: 2^4 x 31

Step 2: Now we have found the prime factors of 496. The second step is to make pairs of these prime factors. Since 496 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √496 using prime factorization alone is not straightforward.

The long division method is particularly useful 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 496, we group it as 96 and 4.

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

Step 3: Now bring down 96, making it 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 the dividend and quotient. We now have 4n as the new divisor and need to find the value of n.

Step 5: Find 4n x n ≤ 96; let n be 2, now 4 x 2 x 2 = 16.

Step 6: Subtract 96 from 16 to get 80, and the quotient is 22.

Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to bring down two zeros to the dividend. Now the new dividend is 8000.

Step 8: The new divisor is 44 because 442 x 2 = 884.

Step 9: Subtracting 884 from 8000 gives 7116 as the result.

Step 10: Now the quotient is approximately 22.27.

Step 11: Continue these steps until you get two numbers after the decimal point. If there is no decimal value, continue until the remainder is zero.

So the square root of √496 is approximately 22.27.

The approximation method is another method for finding square roots; it is an easy way to find the square root of a given number. Now let us learn how to find the square root of 496 using the approximation method.

Step 1: Find the closest perfect squares to √496. The smallest perfect square less than 496 is 484, and the largest perfect square greater than 496 is 529. √496 falls somewhere between 22 and 23.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (496 - 484) / (529 - 484) = 12 / 45 ≈ 0.27. Adding this to the integer part gives us 22 + 0.27 = 22.27.

So the square root of 496 is approximately 22.27.

• Square root: A square root is the inverse of a square. For example, 4^2 = 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 factorization: The process of expressing a number as a product of its prime factors.
   
• Approximation method: A method used to find an approximate value of a square root or other mathematical expression.
   
• Long division method: A step-by-step method used to find the square root of a non-perfect square number.