Question

Using a number line, what is the approximate value of
19−−√?

Answer 1

To find the approximate value of \( \sqrt{19} \) using a number line, we can estimate where \( \sqrt{19} \) lies between two perfect squares.

1. **Identify perfect squares around 19**:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)

This means \( \sqrt{19} \) will be between 4 and 5:

\[
4 < \sqrt{19} < 5
\]

2. **Estimate further**:
Next, we can check values between 4 and 5 to find a more precise estimate:
- \( 4.4^2 = 19.36 \) (too high)
- \( 4.3^2 = 18.49 \) (too low)

Since \( 4.3^2 < 19 < 4.4^2 \), we know:

\[
4.3 < \sqrt{19} < 4.4
\]

3. **Narrowing it down**:
Checking a value between 4.3 and 4.4:
- \( 4.35^2 = 18.9225 \) (too low)
- \( 4.36^2 = 19.0496 \) (too high)

Now we have:

\[
4.35 < \sqrt{19} < 4.36
\]

4. **Final approximation**:
Based on the previous checks and estimates, we can approximate \( \sqrt{19} \) to be about \( 4.36 \).

So, the approximate value of \( \sqrt{19} \) is about **4.36**.