What is the product?
[tex]$\left(3 a^2 b^4\right)\left(-8 a b^3\right)$[/tex]
Answer (2)
The product of the expression ( 3 a 2 b 4 ) ( − 8 a b 3 ) is − 24 a 3 b 7 . This is found by multiplying the coefficients and combining the exponents of like bases using the power rule. Therefore, we conclude with − 24 a 3 b 7 .
;
Multiply the coefficients: 3 × − 8 = − 24 .
Multiply the 'a' terms: a 2 × a = a 2 + 1 = a 3 .
Multiply the 'b' terms: b 4 × b 3 = b 4 + 3 = b 7 .
Combine the results: − 24 a 3 b 7 .
Explanation
Understanding the Problem We are asked to find the product of the expression ( 3 a 2 b 4 ) ( − 8 a b 3 ) . This involves multiplying the coefficients and adding the exponents of like variables.
Multiplying Coefficients First, we multiply the coefficients: 3 × − 8 = − 24 .
Multiplying 'a' Terms Next, we multiply the a terms: a 2 × a = a 2 + 1 = a 3 .
Multiplying 'b' Terms Then, we multiply the b terms: b 4 × b 3 = b 4 + 3 = b 7 .
Combining the Results Finally, we combine the results to get the product: − 24 a 3 b 7 .
Examples
Understanding how to multiply expressions with variables and exponents is crucial in many fields. For example, in physics, when calculating the force between two charged particles, you often deal with expressions involving variables raised to certain powers. Similarly, in computer graphics, transformations of objects in 3D space involve matrix multiplications that require combining variables and their exponents. Mastering these algebraic manipulations allows for accurate modeling and simulations in various scientific and technological applications.
