Simplify $(3 x^3 y^2-5 x y^4-2 x y)+(2 x^3 y^2+5 x y^4+3 x y)$.
Answer (2)
Combine the x 3 y 2 terms: 3 x 3 y 2 + 2 x 3 y 2 = 5 x 3 y 2 .
Combine the x y 4 terms: − 5 x y 4 + 5 x y 4 = 0 .
Combine the x y terms: − 2 x y + 3 x y = x y .
The simplified expression is 5 x 3 y 2 + x y .
Explanation
Understanding the Problem We are given the expression ( 3 x 3 y 2 − 5 x y 4 − 2 x y ) + ( 2 x 3 y 2 + 5 x y 4 + 3 x y ) . Our goal is to simplify it by combining like terms.
Identifying Like Terms First, let's identify the like terms. We have x 3 y 2 terms, x y 4 terms, and x y terms.
Combining x 3 y 2 Terms Now, let's combine the x 3 y 2 terms: 3 x 3 y 2 + 2 x 3 y 2 = ( 3 + 2 ) x 3 y 2 = 5 x 3 y 2 .
Combining x y 4 Terms Next, let's combine the x y 4 terms: − 5 x y 4 + 5 x y 4 = ( − 5 + 5 ) x y 4 = 0 x y 4 = 0 .
Combining x y Terms Finally, let's combine the x y terms: − 2 x y + 3 x y = ( − 2 + 3 ) x y = 1 x y = x y .
Final Simplified Expression Now, let's put it all together. The simplified expression is 5 x 3 y 2 + 0 + x y = 5 x 3 y 2 + x y .
Conclusion Therefore, the simplified expression is 5 x 3 y 2 + x y .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics and has practical applications in various fields. For instance, consider a scenario where you are calculating the area of a rectangular garden with variable side lengths. If the length is represented by 3 x 3 y 2 − 5 x y 4 − 2 x y and the width is represented by 2 x 3 y 2 + 5 x y 4 + 3 x y , the total area can be found by multiplying these expressions. Simplifying the expressions first, as we did in this problem, makes the subsequent calculations easier and more manageable. This skill is also crucial in physics, engineering, and computer science for modeling and solving complex problems.
To simplify the expression, we combine like terms leading to 5 x 3 y 2 + x y as the final result. This involves adding the coefficients of each type of term. The simplified expression is 5 x 3 y 2 + x y .
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