What is the Square Root of 11?
Rounding to eight decimal places, the square root of 11 is 3.31662479.
The square root of 11 can be written in following ways:
Square root of 11 in radical form: \(\sqrt{11}\)
Square Root of 11 in decimal form: 3.31662479……
Square root of 11 in exponent form: \((11)^{\frac{1}{2}}\) or \((11)^{0.5}\).
Square Root of 11 by Long Division Method
Since, 11 is not a perfect square so its square root will not be an integer.
Step 1 : Write 11 in decimal form such that a pair of zeros should be there after decimals. Write 11 as 11.00 00 00 00
Step 2: Since \(3^2=9\) and \(4^2=16\), so the square root of 11 will lie between 3 and 4. Find a number that, when multiplied to itself, gives the product less than or equal to 11 .Use the lower number to divide 11 like 3 \(\times\) 3 = 9 and 11 – 9 = 2 .
Step 3: Draw a pair of zero in carry over that makes 2 as 200. Add both the three of divisor and quotient, 3 + 3 = 6 . Now the second divisor will be a number of which 6 will be at tens palace and identify a number in unit place such that when multiplied to itself, gives the product less than or equal to 200. 63 \(\times\) 3 = 189 and 200 – 189 = 11
Step 4 : Draw a pair of zero in carry over that makes 11 as 1100.
Find the next divisor of 1100. The tens and hundred places will be 63 + 3 = 66 and identify a number in the unit place such that when multiplied to itself, gives the product less than or equal to 1100. keep Repeating step 4.
Finally, we estimate \(\sqrt{11}\) = 3.3166.
Square root of 11 by Prime Factorization Method
- In this method, we need to find the prime factorization of 11.
- Pair the repeated factors.
- Take out the one factor from each pair.
11 = 11 \(\times\) 1
\(\sqrt{11}=\sqrt{11 \times 1}=\sqrt{11}\times \sqrt{1}\)
Square root of 11 is \(\sqrt{11}\) ≈ 3.317.
The unpaired prime factors which remains under the radical sign (‘ \(\sqrt{}\) ’), use their approximate square root value from the list given below:
Square root of 2 is \(\sqrt{2}\) ≈ 1.414
Square root of 3 is \(\sqrt{3}\) ≈ 1.732
Square root of 5 is \(\sqrt{5}\) ≈ 2.236
Square root of 7 is \(\sqrt{7}\) ≈ 2.646
Square root of 11 is \(\sqrt{11}\) ≈ 3.317
Square root of 13 is \(\sqrt{13}\) ≈ 3.61
Square root of 17 is \(\sqrt{17}\) ≈ 4.123
Square root of 19 is \(\sqrt{11}\) ≈ 4.359
Solved Square Root of 11 Examples
Example 1: Find the square root of 11 by trial and error method.
Solution:
This method is based on approximation and estimation. We find the square root of the number by guessing the values.
Step 1: 11 lies between two perfect squares 9 and 16.
Step 2: Since \(3^2=9\) and \(4^2=16\) , so the square root of 11 will lie between 3 and 4.
Step 3: Guess the square of a number ‘lower than 4 and greater than 3’. To do this, guess a small random number that will be added to 3 and reduced from 4. Let this be 0.2.
4 – 0.2 = 3.8 3 + 0.2 = 3.2
\(3.8^2\) = 14.44 \(3.2^2\) = 10.24
Let 3.8 – 0.1 = 3.7 Let 3.2 + 0.1 = 3.3
\(3.7^2\) = 13.69 \(3.3^2\) = 10.89
\(3.6^2\) = 12.96 \(3.4^2\) = 11.56
\(3.5^2\) = 12.25 \(3.5^2\) = 12.25
\(3.3^2\) = 10.89 is nearest to 11.
So, we can estimate \(\sqrt{11}\) = 3.3
Example 2: Find the square root of 66.
Solution :
Find prime factorization of 66.
\(\sqrt{66}=\sqrt{2\times 3\times 11}\)
= \(\sqrt{2}\times \sqrt{3} \times \sqrt{11}\)
= 1.414 \(\times\) 1.732 \(\times\) 3.317
= 8.123
Hence, the square root of 66 is 8.123.
Example 3 : Evaluate each expression.
(a) 9\(\sqrt{121}\) + 10
(b) 20\(\sqrt{22}\) + 8
Solution:
(a) 9\(\sqrt{121}\) + 10
= 9 \(\times\) 11 + 10 [Evaluate the square root ,\(11^2\) = 121]
= 99 + 10 = 109
(b) 20\(\sqrt{22}\) + 8
= 20\(\times\sqrt{2\times 11}\) + 8
= 20\(\times\sqrt{2}\times\sqrt{11}\) + 8
= 20 \(\times\) 1.414 \(\times\) 3.317 + 8 [Here, \(\sqrt{2}\) ≈ 1.414 and \(\sqrt{11}\) ≈ 3.317 ]
= 93.805 + 8 = 101.805
Example 4: The area of a square crop field is 121\(ft^2\). Find the side length of the field.
Solution :
Let the side of the square crop field be ‘a’ ft.
The area of a square field = \(a^2\)
121 = \(a^2\) [Given is the area of the square = 121 \(ft^2\)]
\(\sqrt{121}\) = a [Evaluate the square root,\(11^2\) = 121]
11 = a
Hence, the side of the square field is 11 ft.
The square root of a negative number is imaginary. So, by using the concept of complex numbers we can find the square root of a negative number.
A number is said to be a perfect square if the square root of the given number is an integer.
Example: 100 is a perfect square because the square root of 100 is 10. 64 is also a perfect square since the square root of 64 is 8.
Yes , the product of two perfect squares is always a perfect square.
Let’s consider 121 and 64 as perfect squares. The product of these squares is 7744, which is the square of 88.
The measure of length is always positive. So, the negative square root is neglected. So we use only 11 as the measure of side length.