Simplify the following: $-3 x^4 y^3\left(2 x^2 y^2-3 x^4 y^3+4 x y\right)$
Answer (2)
Distribute the term − 3 x 4 y 3 to each term inside the parentheses: − 3 x 4 y 3 ( 2 x 2 y 2 ) − 3 x 4 y 3 ( − 3 x 4 y 3 ) − 3 x 4 y 3 ( 4 x y ) .
Simplify each term: − 6 x 6 y 5 + 9 x 8 y 6 − 12 x 5 y 4 .
Rearrange the terms: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4 .
The simplified expression is: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4 .
Explanation
Understanding the Problem We are given the expression − 3 x 4 y 3 ( 2 x 2 y 2 − 3 x 4 y 3 + 4 x y ) and asked to simplify it. Our strategy will be to distribute the term − 3 x 4 y 3 to each term inside the parentheses and then combine like terms, if any.
Distributing the Term First, we distribute − 3 x 4 y 3 to each term inside the parentheses: − 3 x 4 y 3 ( 2 x 2 y 2 ) − 3 x 4 y 3 ( − 3 x 4 y 3 ) − 3 x 4 y 3 ( 4 x y ) This simplifies to: − 6 x 6 y 5 + 9 x 8 y 6 − 12 x 5 y 4
Rearranging the Terms Now, let's rearrange the terms to match the order of the given expression x 6 y 5 + x 8 y 6 − x 5 y 4 . The simplified expression is: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4
Final Answer Therefore, the simplified form of the given expression − 3 x 4 y 3 ( 2 x 2 y 2 − 3 x 4 y 3 + 4 x y ) is: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4
Examples
Understanding how to simplify algebraic expressions like this is crucial in many areas, such as physics and engineering, where you often need to manipulate equations to solve for unknown variables. For example, if you're calculating the volume of a complex shape or analyzing forces in a mechanical system, you'll frequently encounter expressions that need simplification. Mastering these skills allows for efficient problem-solving and accurate results in real-world applications.
To simplify − 3 x 4 y 3 ( 2 x 2 y 2 − 3 x 4 y 3 + 4 x y ) , we distribute − 3 x 4 y 3 to each term in the parentheses, resulting in the simplified expression 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4 .
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