Choose the correct option:

1. The rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is:



Answer :

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].