Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero)simplify: \\(\frac{2(f^4)^3 f^3}{6f^9}\\)your answer should contain only positive exponents.\\(\frac{f^3}{3}\\)\\(\frac{1}{6f^3}\\)\\(\frac{f^6}{3}\\)\\(\frac{f^{15}}{3}\\)

Explanation:

Step1: Simplify the power of a power

Using the rule \((a^m)^n = a^{mn}\), we simplify \((f^4)^3\) as \(f^{4\times3}=f^{12}\). So the numerator becomes \(2\times f^{12}\times f^3\).
\(2(f^4)^3f^3 = 2f^{12}f^3\)

Step2: Simplify the product of powers

Using the rule \(a^m\times a^n = a^{m + n}\), we add the exponents of \(f\) in the numerator: \(f^{12}\times f^3=f^{12 + 3}=f^{15}\). So the numerator is now \(2f^{15}\) and the expression is \(\frac{2f^{15}}{6f^9}\).

Step3: Simplify the fraction of coefficients and powers

First, simplify the coefficient: \(\frac{2}{6}=\frac{1}{3}\). Then, using the rule \(\frac{a^m}{a^n}=a^{m - n}\) for \(a = f\), \(m = 15\) and \(n = 9\), we get \(f^{15-9}=f^6\). Combining these, we have \(\frac{1}{3}\times f^6=\frac{f^6}{3}\).

Answer:

\(\frac{f^6}{3}\) (corresponding to the option with \(\frac{f^6}{3}\))