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To solve the given expression:

\[
\left( -\frac{1}{3} + 0.6 \left( -\frac{1}{2} \right) + 1 \right)^2
\]

Let's break down the steps:

### Step 1: Simplify Inside the Parentheses
1. Start with simplifying \(0.6 \left(-\frac{1}{2}\right)\):
   \[
   0.6 \times -\frac{1}{2} = -0.3
   \]

2. Substitute this back into the expression:
   \[
   \left( -\frac{1}{3} - 0.3 + 1 \right)^2
   \]

3. Convert \(-0.3\) into a fraction for easier calculation:
   \[
   -0.3 = -\frac{3}{10}
   \]

4. Convert \(-\frac{1}{3}\) and \(-\frac{3}{10}\) to a common denominator (which is 30):
   \[
   -\frac{1}{3} = -\frac{10}{30} \quad \text{and} \quad -\frac{3}{10} = -\frac{9}{30}
   \]

5. Combine these terms with 1 (as \(\frac{30}{30}\)):
   \[
   \left( -\frac{10}{30} - \frac{9}{30} + \frac{30}{30} \right)^2
   \]

### Step 2: Calculate the Sum
1. Add the fractions inside the parentheses:
   \[
   -\frac{10}{30} - \frac{9}{30} + \frac{30}{30} = \frac{11}{30}
   \]

2. The expression now becomes:
   \[
   \left( \frac{11}{30} \right)^2
   \]

### Step 3: Square the Fraction
1. Square \(\frac{11}{30}\):
   \[
   \left( \frac{11}{30} \right)^2 = \frac{121}{900}
   \]

### Final Answer:
\[
\frac{121}{900}
\]

Would you like more details or have any questions about the steps? 

Here are some related questions for further practice:
1. How do you convert a decimal to a fraction?
2. What is the process for adding fractions with different denominators?
3. How do you find the square of a fraction?
4. What is the distributive property, and how is it used in algebraic expressions?
5. How can you simplify expressions with nested parentheses?

**Tip:** When adding fractions, always look for a common denominator to make the addition easier.

Evaluate. (-1/3 + 0.6(-1/2) + 1)^2. What is the value of the expression?

Learn how to evaluate the expression (-1/3 + 0.6(-1/2) + 1)^2 with a step-by-step breakdown. This problem involves arithmetic operations, fraction addition, and squaring, suitable for Grades 7-9.

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