1. **Stating the problem:** We have several expressions and need to identify which are designations (non-propositional expressions) and which are propositions (statements that can be true or false). Then, we determine the truth value (logical value) of the propositions. 2. **Definitions:** - A **proposition** is a statement that is either true or false. - A **designation** is an expression that does not have a truth value (e.g., a name, a phrase without truth value). 3. **Analyze each expression:** - A: $3 + 2 \times (-4)$ is a numerical expression (designation). - B: $\sqrt{2}^3 - 4 = 4$ is a statement that can be true or false (proposition). - C: "Miguel" is a name, not a proposition (designation). - D: "A ilha das Flores pertence ao arquipélago dos Açores" is a statement that can be true or false (proposition). - E: "Paris é capital de França" is a statement that can be true or false (proposition). - F: $\sqrt[3]{-64} = -4$ is a statement that can be true or false (proposition). 4. **Determine truth values:** - B: Calculate $\sqrt{2}^3 - 4$: $$\sqrt{2}^3 = (\sqrt{2})^3 = (2^{1/2})^3 = 2^{3/2} = 2^{1 + 1/2} = 2 \times \sqrt{2} \approx 2 \times 1.414 = 2.828$$ So, $$2.828 - 4 = -1.172 \neq 4$$ Therefore, B is **false**. - D: "A ilha das Flores pertence ao arquipélago dos Açores" is **true** (the island Flores is part of the Azores archipelago). - E: "Paris é capital de França" is **true**. - F: $\sqrt[3]{-64} = -4$: Since $(-4)^3 = -64$, this is **true**. **Final answers:** - Designations: A, C - Propositions: B (false), D (true), E (true), F (true)