To find the product of the given expression [tex]\((3a^2b^7)(5a^3b^8)\)[/tex], we can follow these steps:
1. Identify the coefficients:
- The first term has a coefficient of 3.
- The second term has a coefficient of 5.
2. Multiply the coefficients:
[tex]\[
3 \cdot 5 = 15
\][/tex]
3. Identify the exponents for [tex]\(a\)[/tex]:
- The first term has [tex]\(a^2\)[/tex].
- The second term has [tex]\(a^3\)[/tex].
4. Add the exponents for [tex]\(a\)[/tex]:
[tex]\[
2 + 3 = 5
\][/tex]
5. Identify the exponents for [tex]\(b\)[/tex]:
- The first term has [tex]\(b^7\)[/tex].
- The second term has [tex]\(b^8\)[/tex].
6. Add the exponents for [tex]\(b\)[/tex]:
[tex]\[
7 + 8 = 15
\][/tex]
7. Combine the results:
Substituting the multiplied coefficient and the summed exponents into the expression, we get:
[tex]\[
15 \cdot a^5 \cdot b^{15}
\][/tex]
Now, we compare this resulting expression with the given choices:
- [tex]\(8a^5b^{15}\)[/tex]
- [tex]\(8a^6b^{56}\)[/tex]
- [tex]\(15a^5e^{15}\)[/tex]
- [tex]\(15a^5b^{56}\)[/tex]
The correct choice that matches our resulting expression [tex]\(15a^5b^{15}\)[/tex] is not directly listed.
However, the correct product based on our calculations and the given options can be:
- [tex]\(\boxed{15 a^5 b^{15}}\)[/tex]
The correct simplification of [tex]\((3a^2b^7)(5a^3b^8)\)[/tex] results in the expression [tex]\(\color{red}{15 a^5 b^{15}}\)[/tex].
Thus, the product is [tex]\( \boxed{15 a^5 b^{15} }\)[/tex].
