Answer:
The correct answer is C
[tex]\frac{12ac^{14}}{b^3}[/tex]
Step-by-step explanation:
We want to simplify
[tex]\frac{(a^2b^4c)^{2}(6a^3b)(2c^5)^3}{4a^6b^{12}c^3}[/tex].
Recall this property of exponents,
[tex](a^m)^n=a^{mn}[/tex]
When we apply this property our expression becomes,
[tex]=\frac{(a^4b^8c^2)(6a^3b)(2^3c^15)}{4a^6b^{12}c^3}[/tex].
We rearrange to get
[tex]=\frac{6\times8\times a^4\times a^3 \times b^8\times b\times c^2\times c^{15}}{4a^6b^{12}c^3}[/tex].
We now apply another property of exponents to simplify the numerator.
According to this property,
[tex]a^m\times a^n=a^{m+n}[/tex]
When we apply this property, the expression will now be,
[tex]=\frac{6\times8\times a^{4+3} \times b^{8+1} c^{2+15}}{4a^6b^{12}c^3}[/tex]
This simplifies to,
[tex]=\frac{48\times a^{7} \times b^{9} c^{17}}{4a^6b^{12}c^3}[/tex]
We again apply this property of exponents,
[tex]\frac{a^m}{a^n}=a^{m-n}[/tex].
When we apply this property, the expression will be,
[tex]=12\times a^{7-6} \times b^{9-12} c^{17-3}[/tex]
[tex]=12\times a^1 \times b^{-3} c^{14}[/tex]
We rewrite this as a positive index to get,
[tex]=\frac{12ac^{14}}{b^3}[/tex]
The correct answer is C
