Square Root of 22
Do you know that 22 divided by 7 results in an irrational number (pi) which is the ratio of the circumference of a circle to its diameter? 22 is considered a special number and the square root of 22 is an irrational number. In this chapter, we will calculate the square root of 22 by the long division method along with solved examples and interactive questions.
Let us take a look at what the square root of 22 is.
 Square Root of 22: √22 = 4.6904
 Square of 22: 22^{2} = 484
What Is the Square Root of 22?
The value of the square root of 22 is 4.69041575982343.This the reason why 22 is called a nonperfect square. A perfect square is expressed as the product of an integer by itself.
Square Root of 22 in Radical Form
Expressing the number inside the radical sign √ is the square root of 22 in the radical form. The square root of 22 is already in the lowest form. 22 can be factored as 2 × 11. Therefore, the square root of 22 in radical form is √22
Is the Square Root of 22 Rational or Irrational?
The square root of 22 is NOT a rational number. There is no integer which when multiplied by itself gives 22. Thus, it is not a perfect square. A number is a rational number if it is the quotient of two integers. However, √22 = 4.6904157982343, which shows that it is a nonterminating decimal. Thus, √22 is an irrational number.
How to Find the Square Root of 22?
We can find the square root of 22 by the following two methods:
 Approximation Method
 Long Division Method
Square Root of 22 by Approximation
We can find the square root of 22 by the method of approximation. Identify the two perfect squares between which 22 lies. 22 lies between 16 and 25. So, the square root of 22 lies between the square root of 16 (lower perfect square) and the square root of 25 (higher perfect square). We know that √16= 4 and √25 = 5. Therefore, the whole number part is 4. To find the decimal part, we use the following formula.
Irrational number Lower perfect square / Bigger perfect squareLower perfect square
= (2216) / (2516)
= 6 / 9
=0.6
The whole number part + the decimal part = 4 + 0.6 (approx) 4.6
Square Root of 22 by Long Division Method
To find the square root of 22 by long division,
 Step 1:Write the pair of digits starting from one's place. Here 22 is the pair to be taken.
 Step 2: Find a divisor 'n' such that n × n results in a product that is less than or equal to 22. We know that 4 × 4 = 16
 Step 3: Find the difference as done in the usual division. (2216= 6). Double the quotient that was calculated.4 × 2 and write it in the new divisor's place. Here it's 8. We leave a blank to create a new divisor.
 Step 4: Here we have no other numbers to divide. So, we keep a decimal point after the dividend 22 and quotient 4. Now, place 2 pairs of zeros after the decimal of the dividend. Bring the pair of zeros down.
 Step 5: Find a divisor 'n' such that n × n results in a product that is less than or equal to 600. Write the next quotient as 6. Now we get our new divisor as 86, as 6 × 86 = 516. Do the division and get the remainder. Here we get 84
 Step 6: Bring the next pair of zeros down.
 Step 7: Double the quotient 46. (2 × 46 = 92). Write it in the new divisor's place and leave a blank next to it.
 Step 8: Find a divisor 'n' such that n × n results in a product that is less than or equal to 8400. We find that 9 × 929= 8361. We can stop here as we have found the quotient up to 2 decimal places.
Thus, √22 = 4.69 by long division. We might proceed further to get the answer to further decimal places.
Explore Square roots using illustrations and interactive examples
Important Notes:
 We can find the actual value of √22= 4.69041575982343 by the long division method.
 The square root of 22 is already in its simplest radical form and cannot be simplified.
Challenging Questions:
 What is the value of √√22?
 Can you think of any quadratic equation which has a root as √22?
Square Root of 22 Solved Examples

Example 1: Sam wants to paint a square wall that has an area of 22 feet^{2}. What is the length of the side of the square wall?
Solution
Area of the square wall = 22 feet^{2}
We know that area of a square is: (side × side) square sq units
Side² = 22
Thus, side = √22
Side= 4.69 feet
Hence, the length of the side of the square wall= 4.69 feet 
Example 2: Help James find the two consecutive numbers between which the square root of 22 lies.
Solution
James knows that the perfect squares nearest to 22 are 16 and 25
The square root of 16 is 4
The square root of 25 is 5
These are the two numbers between which the square root of 22 lies.
Hence, James found that the square root of 22 lies between 16 and 25
FAQs On Square Root of 22
What is the square root of 22?
The square root of 22 is 4.6904.
Why is √22 an irrational number?
A number with decimal expansion as nonterminating and nonrepeating is always an irrational number. So, √22 is an irrational number.
What is the square of 22?
The square of 22 is 484.
Is the square root of 22 an irrational number?
Yes, the square root of 22 is an irrational number.
Is square root of 22 a real number?
Yes, the square root of 22 is a real number.