Square Root of 22.5

The square root is the inverse of the square of the number. 22.5 is not a perfect square. The square root of 22.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √22.5, whereas (22.5)^(1/2) in the exponential form. √22.5 ≈ 4.74342, 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 the 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, we need to group the numbers from right to left. In the case of 22.5, we start with 22 and add decimal numbers as needed.

Step 2: Now we need to find n whose square is less than or equal to 22. We can say n as ‘4’ because 4 × 4 = 16, which is less than 22. Now the quotient is 4, and after subtracting 16 from 22, the remainder is 6.

Step 3: Bring down 50 (from adding a decimal point and two zeros, as 22.5 is a decimal number) to make the new dividend 650.

Step 4: The new divisor will be the sum of the old divisor with itself, 4 + 4 = 8, and add a digit to the quotient to estimate the next divisor.

Step 5: Find a digit n such that 8n × n is less than or equal to 650. Let n be 7, then 87 × 7 = 609.

Step 6: Subtract 609 from 650, the difference is 41, and the quotient is 4.7.

Step 7: Since the remainder is not zero, we add another pair of zeros to get 4100.

Step 8: Repeat the process to find the next decimal until the desired precision is reached.

So, the square root of √22.5 ≈ 4.74342.

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 22.5 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √22.5.

The smallest perfect square less than 22.5 is 16, and the largest perfect square greater than 22.5 is 25.

√22.5 falls somewhere between 4 and 5.

Step 2: Now we need to apply the formula:

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

Using the formula: (22.5 - 16) / (25 - 16) = 6 / 9 = 0.6667.

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 4 + 0.6667 ≈ 4.67, adjusting further we get 4.74342 as the square root of 22.5.

• 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.
   
• 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.
   
• Long division method: A technique used to find the square root of non-perfect squares through a series of divisions.
   
• Approximation method: A technique used to estimate the square root by comparing it to nearby perfect squares.