To solve the given expression [tex]\(\left(-2 x^3\right)\left(9 x^{10} y^6\right)\)[/tex], we can break it down into manageable steps.
1. Coefficient Multiplication:
- We start by multiplying the numerical coefficients [tex]\(-2\)[/tex] and [tex]\(9\)[/tex].
[tex]\[
(-2) \times 9 = -18
\][/tex]
2. Applying Product Rule for Exponents on [tex]\(x\)[/tex] Terms:
- The exponents with the same base [tex]\(x\)[/tex] can be added together when multiplying.
[tex]\[
x^3 \times x^{10} = x^{3+10} = x^{13}
\][/tex]
3. Handling [tex]\(y\)[/tex] Term:
- The [tex]\(y\)[/tex] term does not have a matching base in the other factor, so it remains unchanged.
[tex]\[
y^6
\][/tex]
Combining all these parts together, the simplified expression is:
[tex]\[
-18 \times x^{13} \times y^6
\][/tex]
Therefore, the correct option is:
C. [tex]\(-18 x^{13} y^6\)[/tex]
