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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**.

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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.

If (1/3) and (2/3) are two rational numbers, then (1/3 + 2/3)^2 will be the rational number which:
I. lies between (1/2) and (1)
II. is greater than (2/3)
III. is less than (7/8)
IV. is equal to (1)

Learn how to calculate (1/3 + 2/3)^2 step-by-step. This problem helps students practice addition of fractions and squaring rational numbers, while also reviewing comparisons of rational numbers. Suitable for middle school students in Grades 6-8.

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