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Simplify the expression. Express the answer using a positive exponent.

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.

Learn how to simplify the expression involving exponents and fractions step-by-step. This problem covers key concepts in exponents and fraction simplification. Suitable for high school students learning algebra and exponent rules.

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