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Simplify the expression. Express the answer using positive exponents.

The expression provided is:

\[
\left(\frac{g^{-3} h^2 j^7}{g^0 h^{-4} j}\right)^5
\]

To simplify this expression, follow these steps:

### Step 1: Simplify the expression inside the parentheses.
1. **Simplify the base expression:**
   - For \( g \): \( g^{-3} \) divided by \( g^0 \) is \( g^{-3} \) (since \( g^0 = 1 \)).
   - For \( h \): \( h^2 \) divided by \( h^{-4} \) is \( h^{2 - (-4)} = h^{6} \).
   - For \( j \): \( j^7 \) divided by \( j \) is \( j^{7-1} = j^6 \).

So the simplified expression inside the parentheses is:

\[
\frac{g^{-3} h^6 j^6}{1} = g^{-3} h^6 j^6
\]

### Step 2: Raise the simplified expression to the power of 5.
Apply the power of 5 to each term in the expression:

\[
\left(g^{-3} h^6 j^6\right)^5 = g^{-3 \times 5} h^{6 \times 5} j^{6 \times 5}
\]

This simplifies to:

\[
g^{-15} h^{30} j^{30}
\]

### Step 3: Express the answer using positive exponents.
Convert \( g^{-15} \) to a positive exponent:

\[
\frac{h^{30} j^{30}}{g^{15}}
\]

Thus, the final simplified expression is:

\[
\frac{h^{30} j^{30}}{g^{15}}
\]

### Summary:
The simplified expression, using positive exponents, is \( \frac{h^{30} j^{30}}{g^{15}} \).

Would you like any further details or have any other questions?

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### Related Questions:
1. What are the properties of exponents used in simplifying this expression?
2. How would the expression change if the exponent outside the parentheses were negative?
3. Can you explain the steps to simplify a fraction with variables raised to negative exponents?
4. How would the process differ if one of the variables inside the fraction had a fractional exponent?
5. What are some real-world applications of simplifying expressions like this?

### Tip:
When simplifying expressions with exponents, always carefully apply the exponent rules, especially when dealing with negative exponents, to avoid mistakes.

Learn how to simplify an expression with exponents using positive exponents rule. Step-by-step solution provided for converting negative exponents to positive exponents in algebraic expressions. Suitable for Grades 7-9.

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