Final answer:
Expressions are classified as equivalent to either 8(t+4), 8t +4, or neither by applying the distributive property and properties of operations. 8t + 32 and (8.1) + (8.4) are equivalent to 8(t+4), 4(2t+1) and (8 + 1) + (8 + 4) are equivalent to 8t + 4, and 8t + 12 is considered neither.
Step-by-step explanation:
The student is asked to determine which expressions are equivalent to 8(t+4), 8t +4, or neither. By using properties of operations and distributive property, we can categorize each expression as follows:
- 8t + 32 is equivalent to 8(t+4) because when we apply the distributive property to 8(t+4), we multiply 8 by both t and 4, which gives us 8t + 32.
- 8t + 12 is neither equivalent to 8(t+4) nor to 8t + 4.
- 4(2t+1) expands to 8t + 4 when we distribute 4 across 2t and 1, making it equivalent to 8t + 4.
- The expression 41+4+46 seems to be a typo or irrelevant, so it doesn't fit into any category.
- (8.1) + (8.4) simplifies to 8t + 32, assuming that .1 represents t and .4 is to be translated as a 4, which would make it equivalent to 8(t+4).
- (8 + 1) + (8 + 4) is an expression that simplifies to 8t + 4 if we assume that 1 represents t, so it is equivalent to 8t + 4.
- 8t+4 is of course equivalent to itself, 8t + 4.
- 8(t+4), when distributed, will result in 8t + 32, which makes it equivalent to itself.
To conclude, it is important to recognize the distributive property and properties of operations to identify equivalent expressions.