Insert five irrational numbers between `2sqrt(5)` and `3sqrt(3)`.

Answer

To insert five irrational numbers between \(2\sqrt{5}\) and \(3\sqrt{3}\), we will first calculate the approximate values of these two expressions.

### Step 1: Calculate \(2\sqrt{5}\) and \(3\sqrt{3}\)

1. **Calculate \(2\sqrt{5}\)**:
- \(\sqrt{5} \approx 2.236\)
- Therefore, \(2\sqrt{5} \approx 2 \times 2.236 = 4.472\)

2. **Calculate \(3\sqrt{3}\)**:
- \(\sqrt{3} \approx 1.732\)
- Therefore, \(3\sqrt{3} \approx 3 \times 1.732 = 5.196\)

### Step 2: Identify the range

Now we have:
- \(2\sqrt{5} \approx 4.472\)
- \(3\sqrt{3} \approx 5.196\)

The range between these two values is approximately \(4.472\) to \(5.196\).

### Step 3: Find five irrational numbers in this range

To find five irrational numbers between \(4.472\) and \(5.196\), we can consider numbers that are not perfect squares. Here are five irrational numbers:

1. \(\sqrt{21}\)
2. \(\sqrt{22}\)
3. \(\sqrt{23}\)
4. \(\sqrt{26}\)
5. \(\sqrt{27}\)

### Step 4: Verify that these numbers are irrational

- \(\sqrt{21}\), \(\sqrt{22}\), \(\sqrt{23}\), \(\sqrt{26}\), and \(\sqrt{27}\) are all irrational because they are not perfect squares.

### Final Answer

The five irrational numbers between \(2\sqrt{5}\) and \(3\sqrt{3}\) are:
1. \(\sqrt{21}\)
2. \(\sqrt{22}\)
3. \(\sqrt{23}\)
4. \(\sqrt{26}\)
5. \(\sqrt{27}\)