What is the product of (7x^2y^3) (3x^5y^8)?

1. Identify the terms to be multiplied: The problem asks for the product of two algebraic expressions: $(7x^2y^3)$ and $(3x^5y^8)$.
2. Group the coefficients and the variables: To multiply these terms, we group the numerical coefficients and the variables with the same base separately. This is based on the commutative and associative properties of multiplication.
$$(7 \cdot 3) \cdot (x^2 \cdot x^5) \cdot (y^3 \cdot y^8)$$
3. Multiply the coefficients: Multiply the numerical coefficients together.
$$7 \cdot 3 = 21$$
4. Multiply the variables with the same base: When multiplying variables with the same base, we add their exponents. This is a fundamental rule of exponents: $a^m \cdot a^n = a^{m+n}$.
For the $x$ terms: $$x^2 \cdot x^5 = x^{2+5} = x^7$$
For the $y$ terms: $$y^3 \cdot y^8 = y^{3+8} = y^{11}$$
5. Combine the results: Combine the multiplied coefficients and the simplified variable terms to get the final product.
$$21 \cdot x^7 \cdot y^{11} = 21x^7y^{11}$$