Square Root of 10496

The square root is the inverse operation of squaring a number. 10496 is not a perfect square. The square root of 10496 is expressed in both radical and exponential forms. In radical form, it is expressed as √10496, whereas in exponential form, it is expressed as (10496)^(1/2). √10496 equals approximately 102.455, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

For perfect squares, the prime factorization method is used. For non-perfect squares like 10496, the long division method and approximation method are commonly used. Let us explore 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 break down 10496 into its prime factors:

Step 1: Finding the prime factors of 10496.

Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 41: 2^8 x 41.

Step 2: We found the prime factors of 10496. The second step is to make pairs of those prime factors. Since 10496 is not a perfect square, the digits of the number can’t be grouped into equal pairs completely. Thus, calculating 10496 using prime factorization yields no exact integer square root.

The long division method is used particularly for non-perfect squares. Let's find the square root using this method, step by step:

Step 1: To begin, group the numbers from right to left. For 10496, group it as 96 and 104.

Step 2: Find n whose square is the largest perfect square ≤ 104. Here, n = 10, since 10 x 10 = 100. Subtract 100 from 104 to get a remainder of 4. Bring down 96 to get 496.

Step 3: Double the quotient (10) to get 20, which will be part of the new divisor.

Step 4: Find a digit x such that 20x x x ≤ 496. Here, x = 2, since 202 x 2 = 404.

Step 5: Subtract 404 from 496 to get a remainder of 92.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros. Now the new dividend is 9200.

Step 7: Find x such that 204x x x ≤ 9200. Here, x = 4, since 2044 x 4 = 8176.

Step 8: Subtract 8176 from 9200, leaving a remainder of 1024.

Step 9: The quotient is 102.4. Continue these steps until the desired precision is achieved.

The square root of √10496 is approximately 102.455.

The approximation method is another way to find square roots. Here's how to find the square root of 10496 using this method:

Step 1: Determine the closest perfect squares around 10496. The closest perfect squares are 10404 (102^2) and 10609 (103^2). So, √10496 is between 102 and 103.

Step 2: Use interpolation: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (10496 - 10404) / (10609 - 10404) = 92 / 205 ≈ 0.448 Add this value to the smaller root: 102 + 0.448 = 102.448 Thus, the approximated square root of 10496 is 102.448.

• Square root: A square root is the inverse operation of squaring a number. Example: 5^2 = 25, and thus √25 = 5.
   
• Irrational number: An irrational number cannot be expressed as a fraction p/q, where q is not zero and p and q are integers. Example: √2 is irrational.
   
• Perfect square: A number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.
   
• Prime factorization: The expression of a number as the product of its prime factors. Example: 18 = 2 x 3^2.
   
• Long division method: A technique to find the square root of a number by performing division in a step-by-step manner.