Square Root of 922

The square root is the inverse of the square of the number. 922 is not a perfect square. The square root of 922 is expressed in both radical and exponential forms. In the radical form, it is expressed as √922, whereas in the exponential form it is expressed as (922)^(1/2). √922 ≈ 30.36445, which is an irrational number because it cannot be expressed in the form 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 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 922 is broken down into its prime factors:

Step 1: Finding the prime factors of 922 Breaking it down, we get 2 x 461, where 461 is a prime number.

Step 2: Since 922 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 922 using prime factorization alone is not feasible.

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: Begin by pairing the digits from right to left. In the case of 922, we group it as 9 and 22.

Step 2: Find n whose square is ≤ 9. We can say n is '3' because 3 x 3 = 9. Now, the quotient is 3, and after subtracting, the remainder is 0.

Step 3: Bring down 22, forming the new dividend. Add the old divisor with itself: 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor becomes 6n. We need to find the value of n.

Step 5: Find 6n × n ≤ 22. Consider n = 3, we get 6 x 3 = 18.

Step 6: Subtract 18 from 22; the difference is 4, and the quotient is 30.

Step 7: Since the remainder is less than the divisor, add a decimal point, allowing us to add two zeros to the dividend, making it 400.

Step 8: Find the new divisor, 60n. Choose n = 6, because 606 x 6 = 363.

Step 9: Subtract 363 from 400, resulting in 37.

Step 10: The quotient is 30.3. Continue these steps until two decimal places are achieved.

So the square root of √922 is approximately 30.36.

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 922 using the approximation method.

Step 1: Find the closest perfect squares to √922.

The smallest perfect square below 922 is 900, and the largest perfect square above 922 is 961.

√922 falls between 30 and 31.

Step 2: Apply the formula

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

Using the formula (922 - 900) ÷ (961-900) = 22 ÷ 61 ≈ 0.36.

The next step is adding the initial whole number to the decimal: 30 + 0.36 = 30.36.

Hence, the square root of 922 is approximately 30.36.

• Square root: A square root is the inverse of a square. For example, 4² = 16, and the inverse of the square is the square root, that is, √16 = 4.
   
• Irrational number: An irrational number is one that cannot be written in the form p/q, where q is not equal to zero and p and q are integers.
   
• Long division method: A method used to find the square root of numbers that are not perfect squares, involving grouping and division steps.
   
• Approximation method: A method to estimate the square root of a number using nearby perfect squares to find an approximate value.
   
• Prime factorization: The process of expressing a number as the product of its prime factors, often used to simplify square roots.