To approximate the value of \( \sqrt{19} \), we can look for two perfect squares between which 19 lies.
- The perfect square below 19 is \( 16 \) (since \( 4^2 = 16 \)).
- The perfect square above 19 is \( 25 \) (since \( 5^2 = 25 \)).
This tells us that:
\[ 4 < \sqrt{19} < 5 \]
Next, to narrow it down further, we can check the squares of the decimal values around 4.5:
- \( 4.5^2 = 20.25 \) (which is greater than 19)
- \( 4.4^2 = 19.36 \) (still greater than 19)
- \( 4.3^2 = 18.49 \) (which is less than 19)
So we have:
\[ 4.3 < \sqrt{19} < 4.4 \]
Now we can look for the value of \( \sqrt{19} \) more precisely by checking 4.35:
- \( 4.35^2 = 18.9225 \) (which is less than 19)
- \( 4.36^2 = 19.0496 \) (which is greater than 19)
So now we know:
\[ 4.35 < \sqrt{19} < 4.36 \]
Thus, the approximate value of \( \sqrt{19} \) is closer to:
- \( 4.35 \)
From the provided options:
- 4.25
- 4.38
- 4.13
- 4.5
The value that is closest to \( \sqrt{19} \) is 4.38.
Therefore, the approximate value of \( \sqrt{19} \) is 4.38.
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