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.
Multiply the coefficients and add the exponents of like variables: − 3 x 4 y 3 ( 2 x 2 y 2 ) = − 6 x 6 y 5 , − 3 x 4 y 3 ( − 3 x 4 y 3 ) = 9 x 8 y 6 , and − 3 x 4 y 3 ( 4 x y ) = − 12 x 5 y 4 .
Combine the simplified terms: − 6 x 6 y 5 + 9 x 8 y 6 − 12 x 5 y 4 .
Rearrange the terms to present the final simplified expression: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4 .
Explanation
Understanding the Problem We are asked to simplify the expression − 3 x 4 y 3 ( 2 x 2 y 2 − 3 x 4 y 3 + 4 x y ) . This involves distributing the term − 3 x 4 y 3 to each term inside the parentheses and then simplifying by multiplying coefficients and adding exponents of like variables.
Distributing the Term First, 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
Simplifying and Rearranging Now, we write the simplified expression: − 6 x 6 y 5 + 9 x 8 y 6 − 12 x 5 y 4 Rearranging the terms to match the order in the original expression's answer: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4
Final Answer Therefore, the simplified expression is: 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4
Examples
Simplifying algebraic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, when calculating the area or volume of complex shapes, you often need to simplify expressions to get the final answer. Imagine you are designing a garden with different sections, and you need to calculate the total area to determine how much soil to buy. Simplifying the area expression can make the calculation easier and more accurate. This skill is also crucial in physics, engineering, and computer science for modeling and solving problems.
The expression − 3 x 4 y 3 ( 2 x 2 y 2 − 3 x 4 y 3 + 4 x y ) simplifies to 9 x 8 y 6 − 6 x 6 y 5 − 12 x 5 y 4 by distributing and combining like terms.
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