simplify: $\frac{2(f^{4})^{2}f^{2}}{8f^{12}}$. your answer should contain only positive exponents.

Explanation:

Step1: Simplify $(f^4)^2$

According to power - of - a - power rule $(a^m)^n=a^{mn}$, so $(f^4)^2 = f^{4\times2}=f^8$. The expression becomes $\frac{2f^8f^2}{8f^{12}}$.

Step2: Combine $f^8$ and $f^2$

According to product rule $a^m\times a^n=a^{m + n}$, so $f^8f^2=f^{8 + 2}=f^{10}$. The expression is now $\frac{2f^{10}}{8f^{12}}$.

Step3: Simplify the coefficient and use quotient rule

Simplify $\frac{2}{8}=\frac{1}{4}$, and according to quotient rule $\frac{a^m}{a^n}=a^{m - n}$, so $\frac{f^{10}}{f^{12}}=f^{10-12}=f^{-2}$. The expression is $\frac{1}{4}f^{-2}$.

Step4: Make exponent positive

Using the rule $a^{-n}=\frac{1}{a^n}$, we get $\frac{1}{4f^{2}}$.

Answer:

$\frac{1}{4f^{2}}$