Final answer:
None of the provided options A, B, C, or D show equivalent expressions. The expressions fail to align in operations or exponents to produce equality. Key concepts like powers affecting everything inside parentheses and powers of 10 being added when multiplied are crucial to understand why the expressions are not equivalent.
Step-by-step explanation:
The student is asking which pair of expressions is equivalent. To determine this, we must remember that when terms are raised to a power, that power affects everything inside the parentheses. For example, (27x3)(4x2) would mean each term inside the parentheses is raised to that power, resulting in repeated multiplication of that term. Also, when powers of 10 are multiplied, such as 102 × 103, the exponents are added, giving 102+3 = 105.
In the provided options, we can swiftly see that options A, B, and D do not exhibit equivalent expressions, as the operations and exponents do not align to produce equality. For instance, 213x + 2 does not equal 23x + 12, since the base raised to different exponents does not create equivalent expressions just by adding a constant. Similarly, 3x + 4 does not equal 23x + 8, as the bases and operations do not align.
However, choice C (2(3x + 4)) simplifies to 2× 3x + 2 × 4, which equals 6x + 8. If we then examine the other side of choice C, we have x + 22, which clearly does not match 6x + 8. Therefore, none of the provided options, A, B, C, or D, show equivalent expressions.