Answered

Simplify the following 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]

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]



Answer :

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]