Square Root of 22

The square root is the inverse of the square of the number. 22 is not a perfect square. The square root of 22 is expressed in both radical and exponential form. In the radical form, it is expressed as √22, whereas (22)^(1/2) in the exponential form. √22 ≈ 4.69041575982343, 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:

• 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 22 is broken down into its prime factors:

Step 1: Finding the prime factors of 22 Breaking it down, we get 2 x 11: 2^1 x 11^1

Step 2: Now we found out the prime factors of 22. The second step is to make pairs of those prime factors. Since 22 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 22 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 22, we need to group it as 22.

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 x 4 = 16, which is less than 22. Now the quotient is 4 and the remainder is 22 - 16 = 6.

Step 3: Bring down 00 to make it 600. Add the old divisor with the same number 4 + 4 = 8, which will be part of our new divisor.

Step 4: The new divisor is 80. Now we need to find a digit 'd' such that 80d * d ≤ 600. Here, d = 7, as 807 * 7 = 564, which is less than 600.

Step 5: Subtract 564 from 600, the difference is 36.

Step 6: Bring down 00 to make it 3600. The new divisor will be 847.

Step 7: Continue the process to get the square root up to the desired decimal places.

So the square root of √22 ≈ 4.690.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 22 using the approximation method.

Step 1: Now, we have to find the closest perfect squares of √22.

The smallest perfect square less than 22 is 16 and the largest perfect square greater than 22 is 25.

√22 falls somewhere between 4 and 5.

Step 2: Now we need to apply the formula that is:

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

Applying the formula: (22 - 16) / (25 - 16) = 0.6667

Using the formula, we identified the decimal point of our square root.

The next step is adding the integer part we got initially to the decimal point which gives us 4 + 0.6667 = 4.6667

So, the square root of 22 is approximately 4.67.

• Square Root: A square root is the inverse of a square. For 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, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Long Division Method: A technique used to find the square roots of numbers that are not perfect squares by repeatedly guessing, dividing, and refining the guess.
   
• Approximation Method: A method for estimating the square root of a number by finding the two closest perfect squares and interpolating between them.