Compare the following without evaluating the product:
a. i. `(3)/(5) xx (7)/(9) square (3)/(5)" "ii. (3)/(5) xx (7)/(9) square (7)/(9)`

Answer

To compare the two expressions without evaluating the products, we can analyze the fractions involved in each expression.

### Step 1: Identify the expressions
We have two expressions to compare:
1. \( \frac{3}{5} \times \frac{7}{9} \times \frac{3}{5} \)
2. \( \frac{3}{5} \times \frac{7}{9} \times \frac{7}{9} \)

### Step 2: Rewrite the expressions
Let's rewrite the expressions clearly:
1. \( \frac{3}{5} \times \frac{7}{9} \times \frac{3}{5} = \frac{3}{5} \times \frac{3}{5} \times \frac{7}{9} \)
2. \( \frac{3}{5} \times \frac{7}{9} \times \frac{7}{9} = \frac{3}{5} \times \frac{7}{9} \times \frac{7}{9} \)

### Step 3: Compare the fractions
Now, we can see that both expressions have a common factor of \( \frac{3}{5} \). Thus, we can focus on comparing the remaining parts:
1. For the first expression, we have \( \frac{3}{5} \times \frac{3}{5} \) and \( \frac{7}{9} \).
2. For the second expression, we have \( \frac{3}{5} \) and \( \frac{7}{9} \times \frac{7}{9} \).

### Step 4: Analyze the remaining parts
- In the first expression, the remaining part is \( \frac{3}{5} \).
- In the second expression, the remaining part is \( \frac{7}{9} \times \frac{7}{9} = \frac{49}{81} \).

### Step 5: Compare \( \frac{3}{5} \) and \( \frac{49}{81} \)
To compare \( \frac{3}{5} \) and \( \frac{49}{81} \), we can cross-multiply:
- Cross-multiplying gives us:
- \( 3 \times 81 = 243 \)
- \( 5 \times 49 = 245 \)

Since \( 243 < 245 \), we can conclude that:
\[ \frac{3}{5} < \frac{49}{81} \]

### Step 6: Conclusion
Since \( \frac{3}{5} < \frac{49}{81} \), we can conclude that:
1. \( \frac{3}{5} \times \frac{7}{9} \times \frac{3}{5} < \frac{3}{5} \times \frac{7}{9} \times \frac{7}{9} \)

Thus, the final comparison is:
\[ \frac{3}{5} \times \frac{7}{9} \times \frac{3}{5} < \frac{3}{5} \times \frac{7}{9} \times \frac{7}{9} \]