Find the product. $-8 x^5 y^2 \cdot 6 x^2 y$
Answer (2)
Multiply the coefficients: − 8 \tcdot 6 = − 48 .
Multiply the x terms: x 5 \tcdot x 2 = x 5 + 2 = x 7 .
Multiply the y terms: y 2 \tcdot y = y 2 + 1 = y 3 .
Combine the results: The product is − 48 x 7 y 3 .
Explanation
Understanding the Problem We are asked to find the product of the expression − 8 x 5 y 2 \tcdot 6 x 2 y . This involves multiplying the coefficients and the variables with their respective exponents.
Multiplying Coefficients First, let's multiply the coefficients: − 8 \tcdot 6 = − 48 .
Multiplying Variables Next, we multiply the variables with the same base by adding their exponents. For x , we have x 5 \tcdot x 2 = x 5 + 2 = x 7 . For y , we have y 2 \tcdot y = y 2 + 1 = y 3 .
Combining Results Finally, we combine the results to get the final product: − 48 x 7 y 3 .
Examples
Understanding how to multiply expressions with exponents is crucial in various fields, such as physics and engineering. For instance, when calculating the area of a rectangle with sides expressed as algebraic terms, you need to multiply these terms. Suppose the length of a rectangle is 2 x 2 y and the width is 3 x y 3 . The area would be ( 2 x 2 y ) \tcdot ( 3 x y 3 ) = 6 x 3 y 4 . This skill is also fundamental in simplifying complex equations and solving problems related to growth rates and scaling.
The product of the expression − 8 x 5 y 2 ⋅ 6 x 2 y is − 48 x 7 y 3 . This is calculated by multiplying the coefficients and adding the exponents of like terms. The final result combines these elements into one expression.
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