Sure! Let's simplify the given expression step-by-step:
[tex]\[
\frac{2 \left( \mu^4 \right)^2 f^3}{6 f^4}
\][/tex]
### Step 1: Simplify the numerator
First, let's simplify the numerator [tex]\(\left(\mu^4\right)^2\)[/tex]:
[tex]\[
\left(\mu^4\right)^2 = \mu^{4 \cdot 2} = \mu^8
\][/tex]
So the numerator becomes:
[tex]\[
2 \mu^8 f^3
\][/tex]
### Step 2: Write the simplified numerator in the expression
Now we substitute the simplified numerator back into the original expression:
[tex]\[
\frac{2 \mu^8 f^3}{6 f^4}
\][/tex]
### Step 3: Simplify the coefficient
Next, we simplify the coefficient (the numerical part):
[tex]\[
\frac{2}{6} = \frac{1}{3}
\][/tex]
So now the expression is:
[tex]\[
\frac{1 \cdot \mu^8 \cdot f^3}{3 \cdot f^4} = \frac{\mu^8 f^3}{3 f^4}
\][/tex]
### Step 4: Combine the like terms involving [tex]\(f\)[/tex]
Now we need to simplify the fraction involving [tex]\(f\)[/tex]:
[tex]\[
\frac{f^3}{f^4}
\][/tex]
Using the properties of exponents:
[tex]\[
\frac{f^3}{f^4} = f^{3-4} = f^{-1}
\][/tex]
### Step 5: Substitute back into the expression
Substitute [tex]\( f^{-1} \)[/tex] back into the expression:
[tex]\[
\frac{\mu^8 f^{-1}}{3}
\][/tex]
### Step 6: Rewrite to contain only positive exponents
Finally, since we need the answer with positive exponents, we rewrite [tex]\(f^{-1}\)[/tex] as [tex]\(\frac{1}{f}\)[/tex]:
[tex]\[
\frac{\mu^8}{3 f}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{\mu^8}{3 f}}
\][/tex]
