To find a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex], let's first determine the values of [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex].
1. The value of [tex]\(\sqrt{3}\)[/tex] is approximately [tex]\(1.73205\)[/tex].
2. The value of [tex]\(\sqrt{5}\)[/tex] is approximately [tex]\(2.23607\)[/tex].
Next, we'll find a number that lies between these two values. One method to do so is to calculate their average.
The average of [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] can be computed as:
[tex]\[
\text{Average} = \frac{\sqrt{3} + \sqrt{5}}{2}
\][/tex]
Using the approximations, this becomes:
[tex]\[
\text{Average} = \frac{1.73205 + 2.23607}{2} = 1.98406
\][/tex]
Thus, the average of [tex]\(1.73205\)[/tex] and [tex]\(2.23607\)[/tex] is [tex]\(1.98406\)[/tex].
Since this value itself is a rounded average and hence a rational approximation, we have found a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex]:
The rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is [tex]\(\boxed{1.98406}\)[/tex].
