Square Root of 22.3

The square root is the inverse of the square of the number. 22.3 is not a perfect square. The square root of 22.3 is expressed in both radical and exponential form. In the radical form, it is expressed as √22.3, whereas (22.3)^(1/2) in exponential form. √22.3 ≈ 4.7202, 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:

• Long division method
• Approximation method

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, group the numbers from right to left. In the case of 22.3, consider it as 22 and 3.

Step 2: Find n whose square is less than or equal to 22. We can say n is 4 because 4 × 4 = 16 is less than 22. Now the quotient is 4, after subtracting 16 from 22, the remainder is 6.

Step 3: Bring down 30 (since we need to consider the decimal and add two zeros), making the new dividend 630.

Step 4: The new divisor will be twice the quotient obtained, which is 8. Consider 8n as the new divisor and find n such that 8n × n is less or equal to 630.

Step 5: Consider n as 7, now 87 × 7 = 609.

Step 6: Subtract 609 from 630, the difference is 21, and the quotient is 4.7.

Step 7: Since the remainder is 21 and less than the divisor, add another pair of zeros to the dividend, making it 2100.

Step 8: The new divisor is 94, as 947 × 7 = 6629, which is too large, so use 946 × 6 = 5676.

Step 9: Continue this process to refine the quotient to 4.7202.

The approximation method is another way to find square roots, which is an easy method to find the square root of a given number. Now let us learn how to find the square root of 22.3 using the approximation method.

Step 1: Identify the closest perfect squares around √22.3.

The smallest perfect square less than 22.3 is 16 (4^2) and the largest perfect square more than 22.3 is 25 (5^2).

√22.3 falls between 4 and 5.

Step 2: Apply the formula

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

Using the formula (22.3 - 16) / (25 - 16) = 6.3 / 9 ≈ 0.7.

Adding this to the smaller perfect square root gives us 4 + 0.7 = 4.7, so the approximate square root of 22.3 is 4.7.

• 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, √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction, meaning it cannot be expressed as p/q where q ≠ 0 and p and q are integers.
   
• Decimal: A decimal number is a number that includes a whole number and a fractional part, separated by a decimal point. Examples include 7.86, 8.65, and 9.42.
   
• Approximation: Estimating a value that is close to but not exact. In the context of square roots, an approximation provides a close estimate of the irrational root.
   
• Long division method: A step-by-step method used to divide numbers and find square roots, particularly useful for non-perfect squares.