Let's simplify the expression step by step. The expression given is:

$\left(\frac{ef^3g}{10e^0fg^{-4}}\right)^{-3}$

Step 1: Simplify inside the parentheses.

First, simplify the expression inside the parentheses.

1. Simplify $e^0$: Any number raised to the power of 0 is 1, so $e^0 = 1$.

   The expression becomes: $\frac{ef^3g}{10 \cdot 1 \cdot fg^{-4}} = \frac{ef^3g}{10fg^{-4}}$

2. Simplify the fraction:

   • For $f^3$ divided by $f$, subtract the exponents: $f^{3-1} = f^2$.
   • For $g$ divided by $g^{-4}$, subtract the exponents: $g^{1 - (-4)} = g^{1+4} = g^5$.

   The expression inside the parentheses simplifies to: $\frac{ef^2g^5}{10}$

Step 2: Apply the negative exponent outside the parentheses.

Now we apply the exponent $-3$ to the simplified expression: $\left(\frac{ef^2g^5}{10}\right)^{-3}$

Using the rule $\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}$, the expression becomes: $\frac{10^3}{(ef^2g^5)^3}$

Step 3: Simplify further.

1. Simplify the numerator: $10^3 = 1000$.

2. Simplify the denominator:

   • $e^3$
   • $(f^2)^3 = f^{2 \times 3} = f^6$
   • $(g^5)^3 = g^{5 \times 3} = g^{15}$

   The denominator becomes $e^3f^6g^{15}$.

So, the expression simplifies to: $\frac{1000}{e^3f^6g^{15}}$

Final Answer:

$\boxed{\frac{1000}{e^3f^6g^{15}}}$

This is the expression with positive exponents.

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would the answer change if the exponent were $-2$ instead of $-3$?
2. Can you simplify $\left(\frac{a^2b}{c^3d}\right)^{-2}$ using the same process?
3. What is the general rule for dealing with negative exponents in fractions?
4. How does the result change if $e$ is replaced by a constant value?
5. What happens if the numerator and denominator are swapped before applying the negative exponent?

Tip: When simplifying expressions with exponents, always handle the zero exponent first as it simplifies the problem quickly.