Fill in the boxes with `lt` or `gt` without actually finding the required product.
a. i. `(5)/(9) xx (6)/(7) square (5)/(9)" "ii. (5)/(9) xx (6)/(7) square (6)/(7)`

Answer

To solve the problem of filling in the boxes with `lt` (less than) or `gt` (greater than) without actually calculating the products, we can analyze the fractions involved in each part of the question.

### Step-by-step Solution:

**Part a.i:** Compare \(\frac{5}{9} \times \frac{6}{7}\) with \(\left(\frac{5}{9}\right)^2\).

1. **Calculate the value of \(\frac{5}{9} \times \frac{6}{7}\):**
- The product is \(\frac{5 \times 6}{9 \times 7} = \frac{30}{63}\).
- Simplifying \(\frac{30}{63}\) gives us \(\frac{10}{21}\).

2. **Calculate the value of \(\left(\frac{5}{9}\right)^2\):**
- This is \(\frac{5 \times 5}{9 \times 9} = \frac{25}{81}\).

3. **Compare \(\frac{10}{21}\) and \(\frac{25}{81}\):**
- To compare these fractions, we can find a common denominator or cross-multiply.
- Cross-multiplying gives us:
- \(10 \times 81 = 810\)
- \(25 \times 21 = 525\)
- Since \(810 > 525\), we conclude that \(\frac{10}{21} > \frac{25}{81}\).

Thus, we fill in the box with `gt` (greater than).

**Part a.ii:** Compare \(\frac{5}{9} \times \frac{6}{7}\) with \(\left(\frac{6}{7}\right)^2\).

1. **We already calculated \(\frac{5}{9} \times \frac{6}{7}\) as \(\frac{10}{21}\).**

2. **Calculate the value of \(\left(\frac{6}{7}\right)^2\):**
- This is \(\frac{6 \times 6}{7 \times 7} = \frac{36}{49}\).

3. **Compare \(\frac{10}{21}\) and \(\frac{36}{49}\):**
- Again, we can cross-multiply:
- \(10 \times 49 = 490\)
- \(36 \times 21 = 756\)
- Since \(490 < 756\), we conclude that \(\frac{10}{21} < \frac{36}{49}\).

Thus, we fill in the box with `lt` (less than).

### Final Answers:
- a.i: `gt`
- a.ii: `lt`