Square Root of 109
The square root of 109 is expressed as √109 in the radical form and as (109)^{½} or (109)^{0.5} in the exponent form. The square root of 109 rounded up to 5 decimal places is 10.44031. It is the positive solution of the equation x^{2} = 109.
 Square Root of 109: 10.44030650891055
 Square Root of 109 in exponential form: (109)^{½} or (109)^{0.5}
 Square Root of 109 in radical form: √109
1.  What Is the Square Root of 109? 
2.  Is Square Root of 109 Rational or Irrational? 
3.  How to Find the Square Root of 109? 
4.  Important Notes 
5.  Thinking Out of the Box! 
6.  FAQs on Square Root of 109 
What Is the Square Root of 109?
The square root of 109 is obtained by determining the number whose square gives the original number. Can you think, what would the number be? There is no integer whose square gives 109.
√109 = 10.440
To check this answer, find (10.440)^{2} and we can see that we get a number 108.9936... which is very close to 109.
Is the Square Root of 109 Rational or Irrational?
For any number to be a rational number, it should either be terminating or nonterminating or have a repeating pattern in its decimal part. In the previous section, we saw that: √109 = 10.44030650891055... Clearly, this is nonterminating and the decimal part has no repeating pattern. So it is NOT a rational number. Hence, √109 is an irrational number.
How to Find the Square Root of 109?
To find the square root of 109 using various methods:
 Repeated Subtraction
 Prime Factorization
 Estimation and Approximation
 Long Division
If you want to learn more about each of these methods, click here.
Simplified Radical Form of Square Root of 109
109 is a prime number and thus it has only two factors, 1 and 109. To find the square root of a number, we take one number from each pair of the same numbers from its prime factorization and we multiply them. But the factorization of 109 is 1 × 109 which has no pairs of the same numbers. Thus, the simplest radical form of √109 is √109.
Square Root of 109 by Long Division Method
The square root of 109 can be found using the long division as follows:
 Step 1: We pair digits of 109 starting with a digit at one's place. Put a horizontal bar to indicate pairing.
 Step 2: Now we will find a number which when multiplied to itself gives a product of less than or equal to 1. We know 1 × 1 = 1 which is equal to 1. Thus, the quotient is 1.
 Step 3: Now, we have to bring down 00 and multiply the quotient by 2 which would give us 2. 2 is the starting digit of the new divisor.
 Step 4: 0 is placed at one's place of new divisor because when 20 is multiplied by 0 we get 0. The answer obtained is 9 and we bring the 0s down.
 Step 5: Next, we multiply the quotient 10 is multiplied by 2 which gives 20, which then would become the starting digit of the new divisor.
 Step 6: 4 will be placed at one's place of new divisor because on multiplying 204 by 4 we get 816. The answer obtained is 84 and we bring the 0s down.
 Step 7: The quotient 104 when multiplied by 2 gives 208, which will be the starting digit of the new divisor.
 Step 8: 4 will be placed at one's place of new divisor because on multiplying 2084 by 4 we will get 8336. The answer obtained is 64 and we bring the 0s down.
 Step 9: The quotient 1044 is multiplied by 2 gives 2088, which will be the starting digit of the new divisor.
 Step 10: 0 will be placed at one's place of new divisor because on multiplying 20880 by 0 we will get 0. The answer obtained is 64 and we bring the 0s down.
Explore square roots using illustrations and interactive examples
Important Notes:
 The two closest perfect square numbers between which 109 lies are 100 and 121. So √109 lies between √100 and √121, i.e., √109 lies between 10 and 121.
 By using the prime factorization method for 109, we can see that square root of 109 in the simplest radical form is √109.
Think Tank:
 Are the values of √109 and √109 same?
 Is √109 a real number?
Square Root of 109 Solved Examples

Example 1: Can you find the difference between square root of 109 and square root of 100?
Solution
On finding square root of 109 we get, √109 = 10.440. On finding square root of 100 we get, √100 = 10.
The difference between square root of 109 and square root of 100 is (10.440  10) = 0.440 
Example 2: Can you determine the radius of circle having area 109π square inches?
Solution
Let us assume that the radius of the circle is r inches. Then its area using the formula of area of a circle is πr^{2} square inches. By the given information,
πr^{2} = 109π
r^{2} = 109By taking the square root on both sides, √r^{2}= √109. We know that the square root of r^{2} is r.
By calculating the square root of 109, we get the radius of circle is 10.4 inches (approx). 
Example: If the area of an equilateral triangle is 109√3 in^{2}. Find the length of one of the sides of the triangle.
Solution:
Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a^{2} = 109√3 in^{2}
⇒ a = ±√436 in
Since length can't be negative,
⇒ a = √436 = 2 √109
We know that the square root of 109 is 10.440.
⇒ a = 20.881 in
FAQs on the Square Root of 109
What is the Value of the Square Root of 109?
The square root of 109 is 10.4403.
Why is the Square Root of 109 an Irrational Number?
The number 109 is prime. This implies that the number 109 is pairless and is not in the power of 2. Therefore, the square root of 109 is irrational.
What is the Square of the Square Root of 109?
The square of the square root of 109 is the number 109 itself i.e. (√109)^{2} = (109)^{2/2} = 109.
If the Square Root of 109 is 10.440. Find the Value of the Square Root of 1.09.
Let us represent √1.09 in p/q form i.e. √(109/100) = 1.09/10 = 1.044. Hence, the value of √1.09 = 1.044
Evaluate 14 plus 13 square root 109
The given expression is 14 + 13 √109. We know that the square root of 109 is 10.440. Therefore, 14 + 13 √109 = 14 + 13 × 10.440 = 14 + 135.724 = 149.724
What is the Square Root of 109?
The square root of 109 is an imaginary number. It can be written as √109 = √1 × √109 = i √109 = 10.44i
where i = √1 and it is called the imaginary unit.
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