To solve the given problem, we need to multiply the expressions \(3a^2b^7\) and \(5a^3b^8\). We will do this step-by-step, addressing the coefficients, the \(a\) terms, and the \(b\) terms separately.
### Step-by-Step Solution:
1. Multiply the coefficients:
- The coefficients in the expressions are \(3\) and \(5\).
- The product of the coefficients is:
[tex]\[
3 \times 5 = 15
\][/tex]
2. Multiply the \(a\) terms:
- The exponents of \(a\) in the expressions are \(2\) and \(3\).
- When you multiply terms with the same base, you add their exponents:
[tex]\[
a^2 \times a^3 = a^{2+3} = a^5
\][/tex]
3. Multiply the \(b\) terms:
- The exponents of \(b\) in the expressions are \(7\) and \(8\).
- Similarly, when you multiply terms with the same base, you add their exponents:
[tex]\[
b^7 \times b^8 = b^{7+8} = b^{15}
\][/tex]
### Putting it all together:
Combining the results from each step, the product of the expressions \(3a^2b^7\) and \(5a^3b^8\) is:
[tex]\[
15 a^5 b^{15}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{15 a^5 b^{15}}
\][/tex]
