The problem you've uploaded asks to find the value of the expression involving two rational numbers, $\left( \frac{1}{3} \right)$ and $\left( \frac{2}{3} \right)$, specifically:

$\left( \frac{1}{3} + \frac{2}{3} \right)^2$

Let me solve this step by step:

Step 1: Add the rational numbers

$\frac{1}{3} + \frac{2}{3} = \frac{1 + 2}{3} = \frac{3}{3} = 1$

Step 2: Square the result

$(1)^2 = 1$

Conclusion

The value of $\left( \frac{1}{3} + \frac{2}{3} \right)^2$ is $1$.

Matching with the options provided:

• I. lies between $\frac{1}{2}$ and $1$: This is not true.
• II. is greater than $\frac{2}{3}$: True, since $1 > \frac{2}{3}$.
• III. is less than $\frac{7}{8}$: This is false.
• IV. is equal to $1$: This is true.

So, the correct answer is II and IV.

Would you like further details or have any questions?

Here are five related questions to explore:

1. What are rational numbers, and how are they defined?
2. What is the process of squaring rational numbers?
3. How can we compare fractions and determine their relative sizes?
4. How do we simplify expressions with multiple fractions?
5. What other operations can we perform on rational numbers?

Tip: When adding fractions, always ensure they have the same denominator before adding the numerators.