To solve the given expression [tex]\(\left(a^3 b^{12} c^2\right) \times\left(a^5 c^2\right) \times\left(b^5 c^4\right)^0\)[/tex], follow these steps:
1. Simplify the expression inside each term:
- Notice that [tex]\(\left(b^5 c^4\right)^0\)[/tex] simplifies to 1 because any non-zero number or expression raised to the power of 0 is equal to 1.
Therefore, the expression reduces to:
[tex]\[
(a^3 b^{12} c^2) \times (a^5 c^2) \times 1
\][/tex]
Simplifying further:
[tex]\[
(a^3 b^{12} c^2) \times (a^5 c^2)
\][/tex]
2. Combine like terms:
- For the base [tex]\(a\)[/tex], add the exponents from both terms:
[tex]\[
a^{3+5} = a^8
\][/tex]
- For the base [tex]\(b\)[/tex], since it appears only in the first term and the second term has no [tex]\(b\)[/tex], keep the exponent as it is:
[tex]\[
b^{12}
\][/tex]
- For the base [tex]\(c\)[/tex], add the exponents from both terms:
[tex]\[
c^{2+2} = c^4
\][/tex]
3. Write the final simplified expression:
[tex]\[
a^8 b^{12} c^4
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{a^8 b^{12} c^4}
\][/tex]
Comparing with the options given:
A. [tex]\(a^{15} b^{12} c^4\)[/tex]
B. [tex]\(a^8 b^{17} c^8\)[/tex]
C. [tex]\(a^8 b^{12} c^4\)[/tex]
D. [tex]\(a^{14} b^{15} c^9\)[/tex]
The correct option is:
C. [tex]\(a^8 b^{12} c^4\)[/tex]
