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 can follow these steps:

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

First, we need to find the approximate decimal values of \(2\sqrt{5}\) and \(3\sqrt{3}\).

- \(2\sqrt{5} \approx 2 \times 2.236 = 4.472\)
- \(3\sqrt{3} \approx 3 \times 1.732 = 5.196\)

### Step 2: Identify the range

Now we know that \(2\sqrt{5} \approx 4.472\) and \(3\sqrt{3} \approx 5.196\). We need to find five irrational numbers between these two values.

### Step 3: Find irrational numbers

We can find irrational numbers by taking the square roots of numbers that lie between \(20\) and \(27\) (since \( (2\sqrt{5})^2 = 20 \) and \( (3\sqrt{3})^2 = 27 \)).

The numbers we can choose are:
- \(21\)
- \(22\)
- \(23\)
- \(24\)
- \(25\) (but this is \(5\), which is rational)
- \(26\)

Thus, the irrational numbers we can select are:
1. \(\sqrt{21}\)
2. \(\sqrt{22}\)
3. \(\sqrt{23}\)
4. \(\sqrt{24}\)
5. \(\sqrt{26}\)

### Step 4: Conclusion

Therefore, the five irrational numbers between \(2\sqrt{5}\) and \(3\sqrt{3}\) are:
- \(\sqrt{21}\)
- \(\sqrt{22}\)
- \(\sqrt{23}\)
- \(\sqrt{24}\)
- \(\sqrt{26}\)

---