To simplify the given expression [tex]\(\frac{2\left(f^4\right)^2 f^3}{6 f^9}\)[/tex], follow these detailed steps:
1. Simplify the numerator:
- The numerator is [tex]\(2\left(f^4\right)^2 f^3\)[/tex].
- First, simplify [tex]\(\left(f^4\right)^2\)[/tex]:
[tex]\[
(f^4)^2 = f^{4 \cdot 2} = f^8
\][/tex]
- Now, multiply [tex]\(f^8\)[/tex] by [tex]\(f^3\)[/tex]:
[tex]\[
f^8 \cdot f^3 = f^{8 + 3} = f^{11}
\][/tex]
- Therefore, the numerator simplifies to:
[tex]\[
2 f^{11}
\][/tex]
2. Simplify the fraction:
- The given fraction becomes:
[tex]\[
\frac{2 f^{11}}{6 f^9}
\][/tex]
- Simplify the coefficients [tex]\(\frac{2}{6}\)[/tex]:
[tex]\[
\frac{2}{6} = \frac{1}{3}
\][/tex]
- Simplify the exponents of [tex]\(f\)[/tex]:
[tex]\[
\frac{f^{11}}{f^9} = f^{11 - 9} = f^2
\][/tex]
3. Combine the simplified components:
- The resulting fraction is:
[tex]\[
\frac{1}{3} \cdot f^2 = \frac{f^2}{3}
\][/tex]
Hence, the simplified form of the given expression is [tex]\(\frac{f^2}{3}\)[/tex]. So, the correct answer is:
[tex]\[
\frac{f^2}{3}
\][/tex]
