Which expression is equivalent to $2 a^2 b^2 \cdot 4 b^3 c^2$?
A. $6 a^2 b^3 c^2$
B. $6 a^2 b^6 c^2$
C. $8 a^2 b^5 c^2$
D. $8 a^2 b^6 c^2$
Answer (1)
Multiply the coefficients: 2"."4 = 8 .
Multiply the variables with the same base by adding their exponents: b 2 ⋅ b 3 = b 2 + 3 = b 5 .
Keep the variables that do not have the same base: a 2 and c 2 .
Combine the results: 8 a 2 b 5 c 2 .
Explanation
Understanding the Problem We are given the expression 2 a 2 b 2 "."4 b 3 c 2 and asked to find an equivalent expression from the given options.
The options are: 6 a 2 b 3 c 2 6 a 2 b 6 c 2 8 a 2 b 5 c 2 8 a 2 b 6 c 2
Multiplying Coefficients To find the equivalent expression, we need to multiply the given expression. First, we multiply the coefficients:
2 × 4 = 8
Multiplying Variables Next, we multiply the variables with the same base by adding their exponents:
a 2 remains as a 2 because there is no other term with base a .
b 2 × b 3 = b 2 + 3 = b 5 c 2 remains as c 2 because there is no other term with base c .
Combining Results Now, we combine the results to get the equivalent expression:
8 a 2 b 5 c 2
Final Answer Therefore, the expression equivalent to 2 a 2 b 2 ⋅ 4 b 3 c 2 is 8 a 2 b 5 c 2 .
So, the correct answer is 8 a 2 b 5 c 2 .
Examples
Understanding how to simplify algebraic expressions is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a complex shape, you might need to multiply several algebraic terms together. Simplifying these terms correctly, as we did in this problem, allows for easier and more accurate calculations of physical quantities like volume, area, or force. This skill is also fundamental in computer graphics for scaling and transforming objects in 3D space.
