$a ^m \cdot a ^n= a ^{m+n}$, where a is a real number and m and n are integers. Use the product of powers property to simplify the expression $-2 x^3 y^4 x^2 y^3$.
A. $-2(x y)^{12}$
B. $-2(x y)^{72}$
C. $-2 x^6 y^{12}$
D. $-2 x^5 y^7$
Answer (2)
Group like terms: − 2 ( x 3 x 2 ) ( y 4 y 3 ) .
Apply the product of powers property to the x terms: x 3 x 2 = x 3 + 2 = x 5 .
Apply the product of powers property to the y terms: y 4 y 3 = y 4 + 3 = y 7 .
Combine the simplified terms: − 2 x 5 y 7 .
The simplified expression is − 2 x 5 y 7 .
Explanation
Understanding the problem We are given the expression − 2 x 3 y 4 x 2 y 3 and we want to simplify it using the product of powers property, which states that a m "." a n = a m + n . This property tells us that when we multiply terms with the same base, we can add their exponents.
Grouping like terms First, let's group the like terms together. We have the constant -2, the x terms, and the y terms. So we can rewrite the expression as: − 2 ( x 3 x 2 ) ( y 4 y 3 ) .
Simplifying x terms Now, let's simplify the x terms. We have x 3 x 2 . Using the product of powers property, we add the exponents: x 3 + 2 = x 5 .
Simplifying y terms Next, let's simplify the y terms. We have y 4 y 3 . Using the product of powers property, we add the exponents: y 4 + 3 = y 7 .
Combining the terms Finally, let's combine all the simplified terms together. We have the constant -2, the simplified x term x 5 , and the simplified y term y 7 . So the simplified expression is: − 2 x 5 y 7 .
Final Answer Therefore, the simplified expression is − 2 x 5 y 7 .
Examples
The product of powers property is useful in many real-world scenarios. For example, imagine you are calculating the area of a rectangular garden. If the length of the garden is 2 x 3 y 4 and the width is x 2 y 3 , then the area of the garden is the product of the length and width, which is ( 2 x 3 y 4 ) ( x 2 y 3 ) = 2 x 5 y 7 . This property helps simplify complex expressions in various fields like physics, engineering, and computer science.
The simplified expression for − 2 x 3 y 4 x 2 y 3 using the product of powers property is − 2 x 5 y 7 , which corresponds to option D. We first grouped the terms, then simplified the exponents by adding them. The final result is − 2 x 5 y 7 .
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