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 x² = 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

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.

For any number to be a rational number, it should either be terminating or non-terminating or have a repeating pattern in its decimal part. In the previous section, we saw that: √109 = 10.44030650891055... Clearly, this is non-terminating and the decimal part has no repeating pattern. So it is NOT a rational number. Hence, √109 is an irrational number.

To find the square root of 109 using various methods:

1. Repeated Subtraction 
2. Prime Factorization
3. Estimation and Approximation
4. 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

• Square Root of 5
• Square Root of 108
• Square Root of 128
• Square Root of 49
• Square Root of 100

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?