The expression provided in the image is:

$\left( \frac{-8}{9} \right) + \left( \frac{-2}{3} \times \frac{-4}{\frac{1}{2}} \right)$

Let's break it down step by step.

Step 1: Simplify the multiplication inside the parentheses

We have the multiplication of two fractions:

$\frac{-2}{3} \times \left( \frac{-4}{\frac{1}{2}} \right)$

First, simplify the second fraction:

$\frac{-4}{\frac{1}{2}} = -4 \times 2 = -8$

So, now the expression becomes:

$\frac{-2}{3} \times -8$

Step 2: Multiply the fractions

Now, multiply $\frac{-2}{3}$ by $-8$:

$\frac{-2}{3} \times -8 = \frac{16}{3}$

Step 3: Substitute back into the expression

Now, the expression becomes:

$\left( \frac{-8}{9} \right) + \frac{16}{3}$

Step 4: Add the fractions

To add these fractions, we need a common denominator. The least common denominator of 9 and 3 is 9. So, rewrite $\frac{16}{3}$ as $\frac{48}{9}$:

$\frac{-8}{9} + \frac{48}{9} = \frac{-8 + 48}{9} = \frac{40}{9}$

Final Answer:

The value of the expression is:

$\frac{40}{9}$

Would you like further clarification or a breakdown of any steps?

Here are 5 related questions you might find interesting:

1. How do you convert improper fractions to mixed numbers?
2. Can you explain how to find the least common denominator for adding fractions?
3. What is the rule for multiplying two negative fractions?
4. How would the result change if the original numbers were positive?
5. How can you simplify more complex fractional expressions?

Tip: Always simplify fractions as much as possible before performing operations to make calculations easier!