Last updated on May 26th, 2025
When we multiply a number by itself, we get a number, the number which is multiplied is called the square root. It is a very important and interesting part of mathematics. You must have applied it for measuring each side of a square from the total area.
When we multiply a number by itself we get a number, that number is the square root of 109. The square root of 109 is an irrational number. As we cannot write the number in the form of a ratio. It is denoted by 109 and is approximately equal to 10.4403.
Exponential form : 1091/2 ≅ 10.4403.
Radical Form:√109
10
We can find the square root of a number by using methods like: Prime Factorization; Long Division method; Approximation method and Subtraction method.
The breaking down of a number into smaller numbers is prime factorization. Here, 109 is a prime number, it cannot be broken down into smaller numbers other than 1 and 109. So, from this method we cannot find the exact square root, but we confirm that 109 is not a perfect square.
In this method, we get to find the value of the square root precisely.
Grouping the digits: We start with pairing the digits from the decimal part 109.00
Find the number whose square will be less than or equal to 109 i.e., 10 (since 102 = 100).
Subtract 102 = 100 from 109, which leaves us with 9.
Now we bring down two zeros, which makes it 900
Next double the divisor 10, we get 20. Next we find the largest digit which will be lesser than or equal to 900.
Repeat the steps to get the next decimal places.
So after calculation we get, √109 = 10.4403.
As 102 =100 and 112 =121, the square root of 109 lies between 10 and 11.
Start by guessing 10.4 which is nearest to 10.
10.42 = 108.16 which is too less, keep repeating till we reach the nearest number
Go to the next number 10.4403, 10.44032 = 109 which is close.
So, √109 = 10.4403
The subtraction method includes subtracting consecutive odd numbers from 109 to see how many steps we need to reach zero. However, since 109 is not a perfect square, we cannot exactly reach 0.
109 -1 =108
108-3=105
105-5=100
100-7=93
93-9=84
84 -11 =73
73-13=60
60-15=45
45-17=28
28-19=9
9-21=-12
As we did not get zero, we understand that 109 is not a perfect square.
While learning about square roots, students may likely make mistakes, to avoid them a few mistakes with solutions are given below:
If, x = 10.4403, find the value of x.
If square root of x = 10.4403,
Then,
x= (10.4403)2
Here, the root when shifted to the RHS it becomes the squared power of the number
x=10.4403 × 10.4403
x=109.
So the value of x is 109.
Solve for x if x+5 = 6.
x+5 = 6
x+5=62
Here, the root when shifted to the RHS it becomes the squared power of the number
x+5=36
x=36-5
x=31.
So the value of x is 31
Find the value of 109 +25 .
109 +25 .
First we find the root of each number, which will be,
109 =10.4403
25 =5
Now, we add the roots of both the numbers
10.4403 + 5 =15.4403
We get, 15.4403
Solve: 10/√109
To simplify, 10/√109 We multiply the number in the denominator with the numerator and the denominator, which is called rationalizing.
10/√109 x √109/ √109 = 10√109 /√109 , here when two square roots with the same number are multiplied the roots get canceled (in the denominator), and we are left with the same number, hence √109x √109 = 109.
After rationalizing, we get, 10√109 /√109
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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