To find the square root of 11 and estimate its approximate value for plotting on a number line, we can start with identifying perfect squares around number 11.
First, we know that:
\[ 3^2 = 9 < 11 < 16 = 4^2 \]
This tells us that the square root of 11 is between 3 and 4.
To narrow it down further, we can calculate the squares of numbers between 3 and 4. We can start with 3.3 and 3.4.
Calculating \(3.3^2\):
\[ 3.3^2 = 10.89 \]
Calculating \(3.4^2\):
\[ 3.4^2 = 11.56 \]
Now, we see that:
\[ 10.89 < 11 < 11.56 \]
Thus, we have:
\[ \sqrt{11} \text{ is between } 3.3 \text{ and } 3.4. \]
So, filling in the blanks:
3^2 < 11 < 4^2
√11 is between 3.3 and 3.4
3.4^2 = 11.56
This gives us a clearer comparison for plotting on the number line!