Find four rational numbers between 3/7 and 5/7. Find two irrational numbers between 4.5̅6 and 5.̅1.

### Part 1: Find four rational numbers between \(\frac{3}{7}\) and \(\frac{5}{7}\).

To find rational numbers between two given rational numbers, we can first express them with a common denominator.

We have the fractions \(\frac{3}{7}\) and \(\frac{5}{7}\). These two fractions already have the same denominator, so the difference between them is:

\[
\frac{5}{7} - \frac{3}{7} = \frac{2}{7}
\]

Now, we need to find four rational numbers between \(\frac{3}{7}\) and \(\frac{5}{7}\). A simple way to do this is to divide the interval into smaller parts by adding fractions with the same denominator.

Since the difference is \(\frac{2}{7}\), we can divide this interval into five equal parts, which gives us intervals of size \(\frac{1}{35}\).

Now, we add multiples of \(\frac{1}{35}\) to \(\frac{3}{7}\) to get four rational numbers between \(\frac{3}{7}\) and \(\frac{5}{7}\).

- The first number is:
\[
\frac{3}{7} + \frac{1}{35} = \frac{15}{35} + \frac{1}{35} = \frac{16}{35}
\]

- The second number is:
\[
\frac{3}{7} + 2 \times \frac{1}{35} = \frac{15}{35} + \frac{2}{35} = \frac{17}{35}
\]

- The third number is:
\[
\frac{3}{7} + 3 \times \frac{1}{35} = \frac{15}{35} + \frac{3}{35} = \frac{18}{35}
\]

- The fourth number is:
\[
\frac{3}{7} + 4 \times \frac{1}{35} = \frac{15}{35} + \frac{4}{35} = \frac{19}{35}
\]

Thus, the four rational numbers between \(\frac{3}{7}\) and \(\frac{5}{7}\) are:
\[
\frac{16}{35}, \frac{17}{35}, \frac{18}{35}, \frac{19}{35}
\]

### Part 2: Find two irrational numbers between \(4.5\overline{6}\) and \(5.\overline{1}\).

We are given the repeating decimals \(4.5\overline{6}\) and \(5.\overline{1}\), which represent the numbers \(4.566666\ldots\) and \(5.111111\ldots\), respectively.

To find irrational numbers between these two values, we can consider numbers that are non-repeating and non-terminating decimals.

One approach is to take square roots of non-perfect squares in this interval.

#### First Irrational Number:
Consider the square root of \(21\). The square root of 21 is approximately:
\[
\sqrt{21} \approx 4.582575694
\]
This number is irrational and lies between \(4.5\overline{6} \approx 4.566666\ldots\) and \(5.\overline{1} \approx 5.111111\ldots\).

#### Second Irrational Number:
Consider the square root of \(22\). The square root of 22 is approximately:
\[
\sqrt{22} \approx 4.690415759
\]
This number is irrational and also lies between \(4.5\overline{6} \approx 4.566666\ldots\) and \(5.\overline{1} \approx 5.111111\ldots\).

Thus, two irrational numbers between \(4.5\overline{6}\) and \(5.\overline{1}\) are:
\[
\sqrt{21} \approx 4.582575694 \quad \text{and} \quad \sqrt{22} \approx 4.690415759
\]