To find out between which pair of rational numbers \(\sqrt{38}\) lies, we start by calculating \(\sqrt{38}\).
We know that:
\[ 6^2 = 36 \quad \text{and} \quad 7^2 = 49 \]
This tells us that:
\[ 6 < \sqrt{38} < 7 \]
Next, we can check the squares of numbers between 6.1 and 6.5 to narrow it down further:
\[ 6.1^2 = 37.21 \] \[ 6.2^2 = 38.44 \] \[ 6.3^2 = 39.69 \] \[ 6.4^2 = 40.96 \] \[ 6.5^2 = 42.25 \]
From these calculations, we see:
- \(6.1^2 = 37.21\) which is less than 38.
- \(6.2^2 = 38.44\) which is greater than 38.
This means:
\[ 6.1 < \sqrt{38} < 6.2 \]
Next, we can check the squares of numbers around 6.3:
Between \(6.3\) and \(6.5\):
- \(6.3^2 = 39.69\) (greater than \(38\))
- \(6.4^2 = 40.96\) (greater than \(38\))
Thus, \(\sqrt{38}\) does not lie between \(6.3\) and \(6.5\).
To summarize the findings:
- From the checks, we see that \(\sqrt{38}\) is larger than \(6.1\) but less than \(6.2\).
- It also is not beyond \(6.3\).
Clearly, \(\sqrt{38}\) lies between \(6.1\) and \(6.2\).
So, the correct answer is:
Between 6.1 and 6.2.