- The expression \((x - 4)^3 \cdot 6\) matches "The difference of \(x\) and \(4\), cubed, times \(6\)".
- The expression \((6x)^3 - 4\) matches "\(4\) less than the \(3^{rd}\) power of the product of \(6\) and \(x\)".
- The expression \(6x^3 - 4\) matches "The product of \(x\) cubed and \(6\), minus \(4\)".
- The duplicate expression \((6x)^3 - 4\) corresponds to the remaining description "\(4\) less than \(6x\), cubed".
Explanation
- Analyzing the Problem Expressions and Descriptions
We are asked to match four algebraic expressions with four verbal descriptions. First, let's list the provided expressions:
- \((x - 4)^3 \cdot 6\)
- \((6x)^3 - 4\)
- \(6x^3 - 4\)
- \((6x)^3 - 4\) (which is a repeat of expression 2)
Next, we list the provided verbal descriptions:
AThe difference of \(x\) and \(4\), cubed, times \(6\)
B\(4\) less than the \(3^{rd}\) power of the product of \(6\) and \(x\)
CThe product of \(x\) cubed and \(6\), minus \(4\)
D\(4\) less than \(6x\), cubed (which can be interpreted as \((6x - 4)^3\) or potentially an alternative phrasing for \((6x)^3 - 4\)).
By analyzing the structure of these expressions and descriptions, we can systematically match them.
- Matching Expressions to Descriptions
We map the expressions to the descriptions:
- For expression \((x - 4)^3 \cdot 6\): "The difference of \(x\) and \(4\)" is \((x - 4)\). When this is "cubed" we get \((x - 4)^3\). "Times 6" results in \((x - 4)^3 \cdot 6\). This matches description A.
- For expression \((6x)^3 - 4\): The "product of 6 and x" is \(6x\). The "3rd power of the product" is \((6x)^3\). "4 less than" this value is \((6x)^3 - 4\). This matches description B.
- For expression \(6x^3 - 4\): "x cubed" is \(x^3\). The "product of x cubed and 6" is \(6x^3\). "Minus 4" is \(6x^3 - 4\). This matches description C.
- For description D: "4 less than \(6x\), cubed" could be interpreted as \((6x - 4)^3\). However, since that expression is not present in the tiles, and given that \((6x)^3 - 4\) appears twice, it is highly likely that description D corresponds to the remaining instance of the \((6x)^3 - 4\) expression, perhaps due to a slight ambiguity in the verbal phrasing intended to describe \((6x)^3 - 4\).
- Finalizing the Matches
Based on the analysis, the matches are:
- \((x - 4)^3 \cdot 6 \longleftrightarrow\) The difference of \(x\) and \(4\), cubed, times \(6\)
- \((6x)^3 - 4 \longleftrightarrow\) \(4\) less than the \(3^{rd}\) power of the product of \(6\) and \(x\)
- \(6x^3 - 4 \longleftrightarrow\) The product of \(x\) cubed and \(6\), minus \(4\)
- \((6x)^3 - 4 \longleftrightarrow\) \(4\) less than \(6x\), cubed
The expressions have been successfully mapped based on their logical structure.
Examples
Understanding how to translate word problems into algebraic expressions is a foundational skill in algebra. For example, if you are designing a program to calculate the cost of a product that involves a discount or a tax applied in a specific order (like "3 percent off the total price, then add a shipping fee"), you must translate these English phrases into accurate mathematical equations like \(P_{total} = (P \times 0.97) + S\). This process mirrors the way we matched verbal descriptions to their corresponding algebraic forms in this exercise.