Simplify
a) $\left(3 d^2\right)\left(-9 d^2\right)(-5 d)$
b) $-c^4 \cdot 2 a^4$
Answer (1)
Multiply the coefficients and add the exponents for expression a): ( 3 d 2 ) ( − 9 d 2 ) ( − 5 d ) = 135 d 5 .
Multiply the coefficients for expression b): − c 4 ⋅ 2 a 4 = − 2 a 4 c 4 .
The simplified expression for a) is 135 d 5 .
The simplified expression for b) is − 2 a 4 c 4 .
Explanation
Understanding the Problem We are asked to simplify two expressions involving variables and exponents. We will simplify each expression separately, following the rules of exponents and multiplication.
Simplifying Expression a) For expression a) ( 3 d 2 ) ( − 9 d 2 ) ( − 5 d ) , we multiply the coefficients and add the exponents of the variable d .
Multiplying Coefficients First, multiply the coefficients: 3 × − 9 × − 5 = 135 .
Multiplying Variables Next, multiply the variables: d 2 × d 2 × d = d 2 + 2 + 1 = d 5 .
Final Result for a) Therefore, the simplified expression for a) is 135 d 5 .
Simplifying Expression b) For expression b) − c 4 ⋅ 2 a 4 , we multiply the coefficients. The variables are different, so the expression cannot be simplified further.
Multiplying Coefficients Multiply the coefficients: − 1 × 2 = − 2 .
Combining Variables The variables are c 4 and a 4 . Since the variables are different, we simply write them next to each other.
Final Result for b) Therefore, the simplified expression for b) is − 2 a 4 c 4 .
Examples
Simplifying expressions is a fundamental skill in algebra. For example, if you are calculating the volume of a rectangular prism with sides 3 x , 4 x 2 , and 5 x 3 , you would multiply these expressions together to get 60 x 6 . This skill is also useful in physics when dealing with quantities that are expressed in terms of variables and exponents.
