Square Root of 924

The square root is the inverse of the square of the number. 924 is not a perfect square. The square root of 924 is expressed in both radical and exponential form. In the radical form, it is expressed as √924, whereas 924^(1/2) in the exponential form. √924 ≈ 30.396, 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 like 924, the long division method and approximation method are used. Let us now learn these 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 924 is broken down into its prime factors.

Step 1: Finding the prime factors of 924 Breaking it down, we get 2 x 2 x 3 x 7 x 11: 2^2 x 3^1 x 7^1 x 11^1

Step 2: Now we found out the prime factors of 924. The second step is to make pairs of those prime factors. Since 924 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 924 using prime factorization alone is not straightforward.

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 924, we need to group it as 24 and 9.

Step 2: Now we need to find n whose square is less than or equal to 9. We can say n is 3 because 3^2 is 9, which is equal to 9. Now the quotient is 3, after subtracting 9 from 9, the remainder is 0.

Step 3: Now let us bring down 24, 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 × n ≤ 24. Let us consider n as 4; now 6 × 4 = 24.

Step 6: Subtract 24 from 24, and the difference is 0, so the quotient is 34.

Step 7: Since we have reached a remainder of 0, the square root of 924 is approximately 30.396 when continued further, adding decimal places for precision.

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 924 using the approximation method.

Step 1: Now, we have to find the closest perfect square of √924.

The smallest perfect square less than 924 is 900, and the largest perfect square greater than 924 is 961.

√924 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula:

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

Using the formula, (924 - 900) / (961 - 900) = 24 / 61 ≈ 0.393

Adding the integer part we identified initially to the decimal number: 30 + 0.393 ≈ 30.393, so the square root of 924 is approximately 30.396.

• Square root: A square root is the inverse of a square. Example: 4² = 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 use in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is the process of breaking down a number into its prime factors. For example, the prime factorization of 924 is 2 × 2 × 3 × 7 × 11.
   
• 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.