Multiply the monomials.
$(6 a^4)(-4 a^3) =$
$\square$
Answer (1)
Multiply the coefficients: 6 × − 4 = − 24 .
Multiply the variable parts using the exponent rule: a 4 × a 3 = a 4 + 3 = a 7 .
Combine the results to obtain the final product.
The product of the monomials is − 24 a 7 .
Explanation
Understanding the problem We are asked to multiply two monomials: ( 6 a 4 ) and ( − 4 a 3 ) . To do this, we will multiply their coefficients and then multiply their variable parts using the exponent rules.
Multiplying the coefficients First, let's multiply the coefficients: 6 × − 4 = − 24 .
Multiplying the variable parts Next, we multiply the variable parts. Recall the exponent rule: a m × a n = a m + n . So, a 4 × a 3 = a 4 + 3 = a 7 .
Combining the results Finally, we combine the results to get the product of the monomials: − 24 a 7 .
Examples
Understanding how to multiply monomials is fundamental in various fields, such as physics and engineering, where expressions often involve variables raised to different powers. For instance, when calculating the area of a rectangle with sides 2 x 2 and 3 x 3 , you multiply these monomials to find the area 6 x 5 . Similarly, in physics, when dealing with quantities like velocity and time, which might be expressed as monomials, multiplying them helps determine displacement or other related parameters. This skill is also crucial in computer graphics for scaling and transforming objects, where monomial operations are frequently used to manipulate coordinates and sizes.
