Find the product. $-8 x^5 y^2 \cdot 6 x^2 y$
A. $-8 x^{10} y^2$
B. $-8 x^7 y^3$
C. $-48 x^7 y^3$
D. $-48 x^{10} y^2$
Answer (1)
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: − 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 adding the exponents of like variables.
Multiplying Coefficients First, let's multiply the coefficients: − 8 \tcdot 6 = − 48 .
Multiplying x Terms Next, let's multiply the x terms. When multiplying variables with exponents, we add the exponents: x 5 \tcdot x 2 = x 5 + 2 = x 7 .
Multiplying y Terms Now, let's multiply the y terms: y 2 \tcdot y = y 2 + 1 = y 3 .
Combining the Results Finally, we combine the results to get the product: − 48 x 7 y 3 .
Final Answer Therefore, the product of − 8 x 5 y 2 \tcdot 6 x 2 y is − 48 x 7 y 3 .
Examples
Understanding how to multiply expressions with exponents is crucial in various fields. For example, in physics, when calculating the area or volume of objects that change size proportionally, you'll use similar techniques. Imagine you're designing a rectangular solar panel where the length is 2 x 5 meters and the width is 3 x 2 meters. The total area would be ( 2 x 5 ) \tcdot ( 3 x 2 ) = 6 x 7 square meters. This calculation helps determine the panel's energy output based on its size.
