Square Root of 9500

The square root is the inverse of the square of the number. 9500 is not a perfect square. The square root of 9500 is expressed in both radical and exponential form. In the radical form, it is expressed as √9500, whereas in exponential form it is expressed as (9500)^(1/2). √9500 ≈ 97.4679, 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, the prime factorization method is not used for non-perfect square numbers; instead, 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 9500 is broken down into its prime factors.

Step 1: Finding the prime factors of 9500

Breaking it down, we get 2 x 2 x 5 x 5 x 5 x 19: 2^2 x 5^3 x 19

Step 2: Now we have found the prime factors of 9500. The next step is to make pairs of those prime factors. Since 9500 is not a perfect square, the digits of the number cannot be grouped in perfect pairs. Therefore, calculating √9500 using prime factorization alone is not possible.

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 9500, we need to group it as 00 and 95.

Step 2: Now we need to find n whose square is less than or equal to 95. We can say n is 9 because 9 x 9 = 81, which is less than 95. Now the quotient is 9, and after subtracting 81 from 95, the remainder is 14.

Step 3: Bring down the next pair, which is 00, making the new dividend 1400. Add the old divisor with the same number 9 + 9, which gives us 18, the new divisor.

Step 4: Now, we need to find a digit x for which 18x × x ≤ 1400. Trying x = 7 gives us 187 x 7 = 1309.

Step 5: Subtract 1309 from 1400, resulting in a remainder of 91.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down pairs of zeroes.

Step 7: Continue these steps until you have a sufficiently accurate result.

The square root of 9500 using long division is approximately 97.4679.

The approximation method is another method for finding square roots; it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 9500 using the approximation method.

Step 1: Find the closest perfect squares to 9500. The closest perfect square below 9500 is 9409 (which is 97^2) and above it is 9604 (which is 98^2). So √9500 is between 97 and 98.

Step 2: Use linear interpolation to estimate √9500. Using the formula: (9500 - 9409) / (9604 - 9409) = 0.4725 Adding this value to the lower estimate, we have 97 + 0.4725 = 97.4725.

So the square root of 9500 is approximately 97.4725.

• Square root: A square root is the inverse of squaring a number. For example, 10^2 = 100, and the square root of 100 is √100 = 10.
   
• Irrational number: An irrational number cannot be written as a simple fraction; it has non-repeating, non-terminating decimal expansions.
   
• Linear interpolation: A method to estimate values between two known values, often used to approximate square roots.
   
• Long division method: A step-by-step process to find the square root of a number, particularly useful for non-perfect squares.
   
• Perfect square: A number that is the square of an integer. For example, 144 is a perfect square because it is 12^2.