To find the product of two algebraic expressions, we need to multiply the coefficients and add the exponents of the corresponding variables.
Given the expressions:
[tex]\[
(3a^2b^7) \text{ and } (5a^3b^8)
\][/tex]
### Step-by-Step Solution:
1. Multiply the coefficients:
The coefficients of the expressions are [tex]\(3\)[/tex] and [tex]\(5\)[/tex]. Multiplying these coefficients gives:
[tex]\[
3 \times 5 = 15
\][/tex]
2. Add the exponents of [tex]\(a\)[/tex]:
The exponents of [tex]\(a\)[/tex] in the expressions are [tex]\(2\)[/tex] and [tex]\(3\)[/tex]. Adding these exponents gives:
[tex]\[
2 + 3 = 5
\][/tex]
3. Add the exponents of [tex]\(b\)[/tex]:
The exponents of [tex]\(b\)[/tex] in the expressions are [tex]\(7\)[/tex] and [tex]\(8\)[/tex]. Adding these exponents gives:
[tex]\[
7 + 8 = 15
\][/tex]
Combining these results, the product of the given expressions is:
[tex]\[
15a^5b^{15}
\][/tex]
Therefore, the correct product is:
[tex]\[
\boxed{15a^5b^{15}}
\][/tex]
