Using a number line, what is the approximate value of

Question

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.