Square Root of 912

The square root is the inverse of the square of the number. 912 is not a perfect square. The square root of 912 is expressed in both radical and exponential form. In the radical form, it is expressed as √912, whereas (912)^(1/2) in the exponential form. √912 ≈ 30.198, 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 912 is broken down into its prime factors.

Step 1: Finding the prime factors of 912 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 19: 2^4 x 3^1 x 19^1

Step 2: Now we found out the prime factors of 912. The second step is to make pairs of those prime factors. Since 912 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 912 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 912, we need to group it as 12 and 9.

Step 2: Now we need to find n whose square is 9. We can say n as ‘3’ because 3 x 3 is lesser than or equal to 9. Now the quotient is 3 and after subtracting 9-9 the remainder is 0.

Step 3: Now let us bring down 12, which is the new dividend. Add the old divisor with the same number 3 + 3 we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 12. Let us consider n as 2, now 6 x 2 x 2 = 24.

Step 6: Subtract 12 from 24; the difference is -12, which is incorrect, so we adjust n to 1, making it 6 x 1 x 1 = 6. Subtracting 6 from 12 gives a remainder of 6, and the quotient is 31.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 600.

Step 8: Now we need to find the new divisor that is 61 because 611 x 9 = 549.

Step 9: Subtracting 549 from 600, we get the result 51.

Step 10: Now the quotient is 30.19.

Step 11: 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 √912 is approximately 30.198.

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 912 using the approximation method.

Step 1: Now we have to find the closest perfect square of √912

The smallest perfect square less than 912 is 900 (30^2) and the largest perfect square greater than 912 is 961 (31^2).

√912 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (912 - 900) / (961 - 900) = 12 / 61 ≈ 0.197

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 30 + 0.197 ≈ 30.197, so the square root of 912 is approximately 30.197.

• Square root: A square root is the inverse of a square. 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, the positive square root is more commonly used in real-world applications.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Approximation: The process of finding a value that is close to, but not exactly, the square root of a number.