What is the product? (3a^2b^7)(5a^3b^8)

1. Identify the expression: The problem asks to find the product of two algebraic terms: $(3a^2b^7)$ and $(5a^3b^8)$.
2. Rearrange and group terms: To multiply these terms, we can rearrange and group the coefficients and the variables with the same base together. This is based on the commutative property of multiplication.
$$(3 \cdot 5) \cdot (a^2 \cdot a^3) \cdot (b^7 \cdot b^8)$$
3. Multiply the coefficients: Multiply the numerical coefficients.
$$3 \cdot 5 = 15$$
4. Multiply the 'a' terms: When multiplying terms with the same base, we add their exponents. This is based on the exponent rule $x^m \cdot x^n = x^{m+n}$.
$$a^2 \cdot a^3 = a^{2+3} = a^5$$
5. Multiply the 'b' terms: Similarly, multiply the terms with base 'b' by adding their exponents.
$$b^7 \cdot b^8 = b^{7+8} = b^{15}$$
6. Combine the results: Combine the results from steps 3, 4, and 5 to get the final product.
$$15 \cdot a^5 \cdot b^{15} = 15a^5b^{15}$$