https://r2.erweima.ai/mathbot/image/1727026869811xdzkjwl8.jpg

Let's go through the exercises on the image you uploaded step by step.

### Exercise 1: Completing the table

We need to fill out the missing columns based on the pattern in the first two rows. Let's break down what we have and fill in the rest:

| Fração Irredutível | Fração Decimal | Dízima      | Classificação           |
|--------------------|----------------|-------------|-------------------------|
| 12/5               | 24/10          | 2,4         | Dízima finita           |
| 11/9               | -------        | 1,(2)       | Dízima infinita periódica|
| 9/25               | 0.36           | -------     | Dízima finita           |
| 10/3               | -------        | 3,(3)       | Dízima infinita periódica|
| 13/6               | 2,16(6)        | -------     | Dízima infinita periódica|
| 9/2                | 4,5            | -------     | Dízima finita           |

Explanation for missing values:

1. **11/9**: We know that \(11 \div 9 = 1,2222...\), which is a repeating decimal (dízima infinita periódica), represented as 1,(2).
2. **9/25**: \(9 \div 25 = 0.36\), which terminates (dízima finita).
3. **10/3**: \(10 \div 3 = 3,333...\), so the repeating part is 3,(3) (dízima infinita periódica).
4. **13/6**: \(13 \div 6 = 2,16666...\), so it's 2,(6) and classified as dízima infinita periódica.

### Exercise 2: Explanation

The number on the card is **428/963**. You are asked to explain why this number cannot be represented as a finite decimal (dízima finita).

- A fraction can be written as a finite decimal if, in its simplest form, the denominator (when fully reduced) is only divisible by 2 or 5.
- The prime factorization of 963 is not divisible by only 2 or 5; therefore, **428/963** cannot be represented as a finite decimal.

### Exercise 3: Fractions and Classification

#### 3.1 Writing as irreducible fractions:

Let's break down the numbers in the diagram:
- **7/35** simplifies to **1/5**.
- **85/320** simplifies to **17/64**.
- **27/(2×3²)** simplifies to **27/18**, which further reduces to **3/2**.
- **7/(2×5³)** remains the same as **7/250**.
- **10/75** simplifies to **2/15**.

#### 3.2 Identifying finite decimals:

- A fraction represents a finite decimal if its denominator has only 2 or 5 as prime factors after simplification.
- Checking the simplified forms:
  - **1/5**: Finite decimal (divisible by 5).
  - **17/64**: Finite decimal (divisible by 2).
  - **3/2**: Finite decimal (divisible by 2).
  - **7/250**: Finite decimal (divisible by 2 and 5).
  - **2/15**: Infinite decimal (divisible by 3, not just 2 or 5).

### Exercise 4: Solving the Expressions

#### 4.1 Expression
\[
\frac{1}{3} + \frac{1}{2} \times \left( \frac{4}{3} - \frac{1}{2} \right) - \left( -\frac{1}{3} \right)
\]
Steps:
1. \(\frac{4}{3} - \frac{1}{2}\) → common denominator is 6, giving \(\frac{8}{6} - \frac{3}{6} = \frac{5}{6}\).
2. \(\frac{1}{2} \times \frac{5}{6} = \frac{5}{12}\).
3. Final expression: \(\frac{1}{3} + \frac{5}{12} + \frac{1}{3}\).
4. Combine: \(\frac{4}{12} + \frac{5}{12} + \frac{4}{12} = \frac{13}{12} = 1 \frac{1}{12}\).

#### 4.2 Expression
\[
(5^{-2} + 4^{-2})^{-1} + \left( -\frac{4}{5} \right)^0
\]
Steps:
1. \(5^{-2} = \frac{1}{25}\) and \(4^{-2} = \frac{1}{16}\).
2. Adding: \(\frac{1}{25} + \frac{1}{16} = \frac{16 + 25}{400} = \frac{41}{400}\).
3. Taking the reciprocal: \(\left( \frac{41}{400} \right)^{-1} = \frac{400}{41}\).
4. \(\left( -\frac{4}{5} \right)^0 = 1\), so the final answer is \( \frac{400}{41} + 1 = \frac{441}{41}\).

#### 4.3 Expression
\[
4 - \sqrt[3]{27} + \frac{1}{2} \times \sqrt{64}
\]
Steps:
1. \(\sqrt[3]{27} = 3\).
2. \(\sqrt{64} = 8\), so \(\frac{1}{2} \times 8 = 4\).
3. Final expression: \(4 - 3 + 4 = 5\).

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Would you like further details or clarifications on any of these problems?

Here are five additional related questions for you:
1. How can you determine if a fraction represents a repeating or terminating decimal?
2. What is the process to simplify fractions into their irreducible form?
3. How do you approach solving complex fractions in algebra?
4. Can all fractions be converted into decimal form, and why?
5. What are the steps to solving an equation with exponents and radicals?

**Tip**: For converting fractions to decimals, always check the denominator's prime factors—this helps to determine if the decimal will terminate or repeat!

The user uploaded a math worksheet containing exercises related to rational numbers, decimals, and operations involving powers and roots.

This worksheet contains exercises on rational numbers, fractions, decimals, and algebraic expressions involving powers and roots. Students will practice converting between fractions and decimals, solving algebraic expressions, and applying the properties of exponents. Suitable for Grades 7-9.

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