What is the product?
$\left(3 a^2 b^4\right)\left(-8 a b^3\right)$
A. $-24 a b$
B. $-24 a^2 b^7$
C. $-24 a^2 b^{12}$
D. $-24 a^3 b^7$
Answer (1)
Multiply the coefficients: 3 × − 8 = − 24 .
Multiply the a terms: a 2 × a = a 3 .
Multiply the b terms: b 4 × b 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 terms with the variable a . We have a 2 × a = a 2 + 1 = a 3 . Remember, when multiplying variables with exponents, we add the exponents.
Multiplying 'b' terms Now, we multiply the terms with the variable b . We have b 4 × b 3 = b 4 + 3 = b 7 . Again, we add the exponents.
Combining the Results Finally, we combine all 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 areas, such as calculating the area of a rectangle or the volume of a box. For example, if the length of a rectangle is 3 a 2 b 4 and the width is 8 a b 3 , then the area of the rectangle is ( 3 a 2 b 4 ) ( 8 a b 3 ) = 24 a 3 b 7 . This concept is also used in physics to calculate quantities like kinetic energy or potential energy, where variables represent physical quantities and exponents represent their relationships.
