Multiply and simplify the following expression:
$\left(\frac{8}{5} x y\right)\left(\frac{1}{3} x^2 y\right)\left(-30 x^2 y\right)$
Answer (2)
Multiply the coefficients: 5 8 × 3 1 × ( − 30 ) = − 16 .
Multiply the x terms: x × x 2 × x 2 = x 5 .
Multiply the y terms: y × y × y = y 3 .
Combine the results: − 16 x 5 y 3 .
Explanation
Understanding the Problem We are asked to multiply and simplify the expression ( 5 8 x y ) ( 3 1 x 2 y ) ( − 30 x 2 y ) . To do this, we will multiply the coefficients and combine the variables by adding their exponents.
Multiplying the Coefficients First, let's multiply the coefficients: 5 8 × 3 1 × ( − 30 ) = 5 × 3 8 × 1 × ( − 30 ) = 15 − 240 = − 16.
Multiplying the x Terms Next, let's multiply the x terms: x × x 2 × x 2 = x 1 + 2 + 2 = x 5 .
Multiplying the y Terms Now, let's multiply the y terms: y × y × y = y 1 + 1 + 1 = y 3 .
Combining the Results Finally, let's combine the results: ( 5 8 x y ) ( 3 1 x 2 y ) ( − 30 x 2 y ) = − 16 x 5 y 3 .
Examples
Understanding how to simplify expressions like this is useful in many areas, such as physics and engineering, where you might need to combine different measurements or quantities. For instance, if you're calculating the volume of a complex shape made of different materials, you might need to multiply expressions involving variables like length, width, and height, and then simplify the result to get a final formula. This skill is also fundamental in computer graphics, where transformations and scaling operations often involve multiplying and simplifying algebraic expressions.
The expression ( 5 8 x y ) ( 3 1 x 2 y ) ( − 30 x 2 y ) simplifies to − 16 x 5 y 3 by multiplying the coefficients and combining the variables. The steps include multiplying the coefficients, adding the exponents of like variables, and then combining the results. Therefore, the final answer is − 16 x 5 y 3 .
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