Square Root of 6500

The square root is the inverse operation of squaring a number. 6500 is not a perfect square. The square root of 6500 can be expressed in both radical and exponential forms. In radical form, it is expressed as √6500, and in exponential form as (6500)^(1/2). √6500 ≈ 80.6226, which is an irrational number because it cannot be represented as a fraction p/q, where p and q are integers and q ≠ 0.

For perfect square numbers, the prime factorization method is used. However, for non-perfect square numbers, methods such as long division and approximation are more suitable. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves breaking down a number into its prime factors. Let's break down 6500 into its prime factors:

Step 1: Find the prime factors of 6500. Breaking it down, we get 2 x 2 x 5 x 5 x 13 x 5: 2^2 x 5^3 x 13

Step 2: Pair the prime factors. Since 6500 is not a perfect square, the factors cannot be perfectly paired.

Thus, calculating √6500 using prime factorization alone is not feasible.

The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root using this method:

Step 1: Group the digits of 6500 from right to left as 65 and 00.

Step 2: Find the largest integer n whose square is less than or equal to 65. n is 8 because 8 x 8 = 64.

Step 3: Subtract 64 from 65 to get a remainder of 1, and bring down the next pair of zeros to make it 100.

Step 4: Double the divisor, which is now 16, and determine a new digit to append to the divisor such that the new divisor times this digit is less than or equal to 100.

Step 5: The next digit is 0, so the new divisor is 160, and the new dividend is 100.

Step 6: Subtract 160 x 0 from 100 to get a remainder of 100. Bring down the next pair of zeros to get 10000.

Step 7: Continue this process to find the square root to the desired decimal places.

The square root of 6500 is approximately 80.62.

The approximation method is a simpler way to find square roots, especially when a precise value is not necessary. Here's how to apply it for 6500:

Step 1: Identify the nearest perfect squares. 6400 (80^2) and 6561 (81^2) are the closest perfect squares around 6500.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). For example, (6500 - 6400) / (6561 - 6400) = 100 / 161 = 0.6211. Adding this to 80 gives us approximately 80.62.

Therefore, the square root of 6500 is approximately 80.62.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4 because 4 x 4 = 16.

• Irrational number: An irrational number cannot be expressed as a simple fraction; it has non-repeating, non-terminating decimals.

• Perfect square: A number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

• Long division method: A technique used to find the square root of a non-perfect square by dividing the number into groups of two digits from right to left and performing systematic division.

• Approximation: A method of finding a number close to the exact square root by identifying nearby perfect squares and using interpolation.