What is the product of (3a^2b^4) and (-8ab^3)?

1. Identify the expression: The problem asks for the product of two algebraic terms: $(3a^2b^4)$ and $(-8ab^3)$.
2. Rearrange and group terms: To multiply these terms, we can group the coefficients and the variables separately. This is based on the commutative and associative properties of multiplication.
$$(3a^2b^4) \times (-8ab^3) = (3 \times -8) \times (a^2 \times a) \times (b^4 \times b^3)$$
3. Multiply the coefficients: Multiply the numerical coefficients together.
$$3 \times -8 = -24$$
4. Multiply the 'a' variables: When multiplying variables with exponents, we add the exponents if the bases are the same. Remember that 'a' can be written as $a^1$.
$$a^2 \times a^1 = a^{2+1} = a^3$$
5. Multiply the 'b' variables: Similarly, add the exponents for the 'b' variables.
$$b^4 \times b^3 = b^{4+3} = b^7$$
6. Combine the results: Combine the results from steps 3, 4, and 5 to get the final product.
$$-24 \times a^3 \times b^7 = -24a^3b^7$$