Let's go through each question to find the exact values as requested.

Problem 1:

$\frac{7}{8} + \frac{1}{6} \div \frac{2}{9}$

Step 1: Division

The expression contains a division, so let's handle that first: $\frac{1}{6} \div \frac{2}{9} = \frac{1}{6} \times \frac{9}{2} = \frac{9}{12} = \frac{3}{4}$

Step 2: Addition

Now substitute $\frac{3}{4}$ back into the expression: $\frac{7}{8} + \frac{3}{4}$

Convert $\frac{3}{4}$ to have a common denominator of 8: $\frac{3}{4} = \frac{6}{8}$

Now, add $\frac{7}{8}$ and $\frac{6}{8}$: $\frac{7}{8} + \frac{6}{8} = \frac{13}{8} = 1\frac{5}{8}$

So, the exact value of Problem 1 is: $1\frac{5}{8} \text{ or } \frac{13}{8}$

Problem 2:

$\frac{5}{6} + \frac{2}{3} - \frac{12}{35} \times \frac{7}{9}$

Step 1: Multiplication

First, handle the multiplication: $\frac{12}{35} \times \frac{7}{9} = \frac{12 \times 7}{35 \times 9} = \frac{84}{315}$

Simplify $\frac{84}{315}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 21: $\frac{84}{315} = \frac{4}{15}$

Step 2: Addition and Subtraction

Now, substitute $\frac{4}{15}$ back into the expression: $\frac{5}{6} + \frac{2}{3} - \frac{4}{15}$

Convert each fraction to have a common denominator of 30: $\frac{5}{6} = \frac{25}{30}, \quad \frac{2}{3} = \frac{20}{30}, \quad \frac{4}{15} = \frac{8}{30}$

Now, perform the operations: $\frac{25}{30} + \frac{20}{30} - \frac{8}{30} = \frac{37}{30} = 1\frac{7}{30}$

So, the exact value of Problem 2 is: $1\frac{7}{30} \text{ or } \frac{37}{30}$

Would you like more details on any of these steps or have any questions?

Here are some related questions for further practice:

1. What is the exact value of $\frac{3}{4} + \frac{5}{6}$?
2. How do you simplify the expression $\frac{4}{9} \div \frac{2}{3} + \frac{1}{2}$?
3. Find the value of $\frac{1}{2} + \frac{3}{8} \times \frac{4}{5}$.
4. How do you handle complex expressions with multiple fractions and operations?
5. What is the result of $\frac{6}{7} - \frac{3}{5} \times \frac{10}{11}$?

Tip: When adding or subtracting fractions, always look for a common denominator to make calculations easier.