Square Root of 8500

The square root is the inverse of the square of a number. 8500 is not a perfect square. The square root of 8500 is expressed in both radical and exponential form. In the radical form, it is expressed as √8500, whereas (8500)^(1/2) in the exponential form. √8500 ≈ 92.195, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 8500 is broken down into its prime factors.

Step 1: Finding the prime factors of 8500 Breaking it down, we get 2 x 2 x 5 x 5 x 5 x 17: 2^2 x 5^3 x 17

Step 2: Now we found out the prime factors of 8500. The second step is to make pairs of those prime factors. Since 8500 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating √8500 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 8500, we need to group it as 85 and 00.

Step 2: Now we need to find n whose square is less than or equal to 85. We can say n as ‘9’ because 9 x 9 = 81 is lesser than 85. Now the quotient is 9, subtracting 81 from 85 gives a remainder of 4.

Step 3: Bring down 00, making the new dividend 400. Add the old divisor with the same number: 9 + 9 = 18, which will be our new divisor.

Step 4: The task is to find a number n such that 18n x n ≤ 400. Let us consider n as 2, now 182 x 2 = 364.

Step 5: Subtracting 364 from 400 gives a difference of 36, and the quotient becomes 92.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding a decimal point allows us to add two zeroes to the dividend. Now, the new dividend is 3600.

Step 7: The next step is to find the new divisor, which will be 184, as 184 x 2 = 368.

Step 8: Subtracting 368 from 3600 gives a result of 232, and the quotient becomes 92.1. Continue doing 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 √8500 ≈ 92.195

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 8500 using the approximation method.

Step 1: Identify the closest perfect squares around 8500. The smallest perfect square is 8400, and the largest perfect square is 8649. √8500 falls somewhere between 92 and 93.

Step 2: Apply the formula:

(Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula: (8500 - 8400) ÷ (8649 - 8400) ≈ 0.195

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 92 + 0.195 = 92.195, so the square root of 8500 is approximately 92.195.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is usually the positive square root that is more prominent due to its uses in real-world applications. This is known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 8500 is 2 x 2 x 5 x 5 x 5 x 17.
   
• Long division method: A technique used to find the square root of non-perfect square numbers by dividing and finding the closest perfect square iteratively.