Verified Answer :

  • Group the numerical coefficients together and multiply them: \(3 \times (-8) = -24\).
  • Multiply the variables with the same base using the exponent rule \(x^m \cdot x^n = x^{m+n}\): \(a^2 \cdot a^1 = a^3\) and \(b^4 \cdot b^3 = b^7\).
  • Combine the results to get the final simplified product: \(-24a^3b^7\).
  • The correct option is a, giving the final answer \(\boxed{-24a^3b^7}\).

Explanation

  1. Problem Analysis and Setup

To find the product of the two monomials \((3a^2b^4)\) and \((-8ab^3)\), we need to multiply the coefficients and the corresponding variable terms separately. The given expression is:
\[ (3a^2b^4)(-8ab^3) \]
We can rearrange this using the commutative and associative properties of multiplication:
\[ (3 \times -8) \times (a^2 \times a) \times (b^4 \times b^3) \]
The objective is to compute this product and determine the correct simplified form.

  1. Calculating the Product

Now, we perform the multiplication step-by-step:

  1. Multiply the numerical coefficients:

\[ 3 \times (-8) = -24 \]

  1. Multiply the terms with base \(a\): Using the law of exponents \(x^m \cdot x^n = x^{m+n}\):

\[ a^2 \times a^1 = a^{2+1} = a^3 \]

  1. Multiply the terms with base \(b\): Again using the law of exponents:

\[ b^4 \times b^3 = b^{4+3} = b^7 \]
By combining these results, we get:
\[ -24 \times a^3 \times b^7 = -24a^3b^7 \]
This matches option (a).

  1. Conclusion and Selection of Option

Based on the calculation, the simplified expression for \((3a^2b^4)(-8ab^3)\) is \(-24a^3b^7\). Comparing this result with the given choices:

A\(-24 a^3 b^7\)
B\(-24 a^2 b^{12}\)
C\(-24 a b\)
D\(-24 a^2 b^7\)

The correct option is (a). The final result is \(\boxed{-24a^3b^7}\).

Examples

In physics and engineering, multiplying monomials is common when calculating physical quantities. For instance, if you are determining the force acting on an object where mass depends on a variable \(a\) (like density) and velocity depends on a variable \(b\), you might end up multiplying expressions of this form to find the final kinetic energy or momentum. Mastering these algebraic operations allows for efficient simplification of complex physical formulas.