Square Root of 9375

The square root is the inverse of the square of the number. 9375 is not a perfect square. The square root of 9375 is expressed in both radical and exponential form. In the radical form, it is expressed as √9375, whereas (9375)^(1/2) in the exponential form. √9375 ≈ 96.8653, 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 where 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 9375 is broken down into its prime factors.

Step 1: Finding the prime factors of 9375

Breaking it down, we get 3 x 5 x 5 x 5 x 5 x 5: 3^1 x 5^5.

Step 2: Now we found out the prime factors of 9375. The second step is to make pairs of those prime factors. Since 9375 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 9375 using prime factorization is limited.

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 9375, we need to group it as 75 and 93.

Step 2: Now we need to find n whose square is ≤93. We can say n as ‘9’ because 9 x 9 is less than or equal to 93. Now the quotient is 9; after subtracting 81 from 93, the remainder is 12.

Step 3: Now let us bring down 75, which is the new dividend. Add the old divisor with the same number 9 + 9; we get 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 18n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 18n x n ≤ 1275. Let us consider n as 7. Now, 18 x 7 x 7 = 1764. Since 1764 is greater than 1275, try n as 6.

Step 6: With n as 6, 18 x 6 x 6 = 1296. Subtracting 1296 from 1275 is not possible as 1296 is greater, so n has to be decreased.

Step 7: With n as 5, 18 x 5 x 5 = 1125. Subtract 1125 from 1275; the difference is 150, and the quotient is 95.

Step 8: Now, since the number of digits in the dividend is less than the divisor, we add a decimal point to continue with zeros. Now, the new dividend is 15000.

Step 9: The next divisor is 1950, as 195 x 9 = 1755.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So, the square root of √9375 ≈ 96.87

The approximation method is another method for finding the 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 9375 using the approximation method.

Step 1: Now we have to find the closest perfect square of √9375. The smallest perfect square less than 9375 is 9216, and the largest perfect square greater than 9375 is 9604. √9375 falls somewhere between 96 and 97.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (9375 - 9216) ÷ (9604 - 9216) = 159 ÷ 388 ≈ 0.41

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 96 + 0.41 = 96.41, so the square root of 9375 is approximately 96.41.

• 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. This is why it is also known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its prime factors. Example: The prime factorization of 18 is 2 x 3 x 3.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.