Simplify
a) $\left(3 d^2\right)\left(-9 d^2\right)(-5 d)$
Answer (1)
Multiply the coefficients: 3 × − 9 × − 5 = 135 .
Multiply the variable terms: d 2 × d 2 × d = d 2 + 2 + 1 = d 5 .
Combine the results: 135 d 5 .
The simplified expression is 135 d 5 .
Explanation
Understanding the problem We are asked to simplify the expression ( 3 d 2 ) ( − 9 d 2 ) ( − 5 d ) . This involves multiplying the coefficients and combining the powers of the variable d .
Multiplying the coefficients First, let's multiply the coefficients: 3 × − 9 × − 5 = 135 .
Multiplying the variable terms Next, let's multiply the variable terms: d 2 × d 2 × d = d 2 + 2 + 1 = d 5 .
Combining the results Finally, combine the results to get the simplified expression: 135 d 5 .
Final Answer Therefore, the simplified expression is 135 d 5 .
Examples
Imagine you are calculating the volume of a rectangular prism where the sides are expressed in terms of d . If the sides are 3 d 2 , − 9 d 2 , and − 5 d , then the volume would be ( 3 d 2 ) ( − 9 d 2 ) ( − 5 d ) = 135 d 5 . Simplifying such expressions is crucial in various engineering and physics applications where polynomial expressions are frequently used to model physical quantities.
