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To solve this problem, we need to simplify the given expression as a power of \( a \).

The expression is:
\[
\frac{\left( a^2 a^4 \right) \div a^{-3}}{a \div \left( a^2 a^3 \right)^4}
\]

Let's go through the steps:

### Step 1: Simplify the Numerator
The numerator is:
\[
\left( a^2 a^4 \right) \div a^{-3}
\]
1. Combine \( a^2 \) and \( a^4 \): \( a^2 \cdot a^4 = a^{2+4} = a^6 \).
2. Now we have \( a^6 \div a^{-3} \).
3. Using the rule \( a^m \div a^n = a^{m - n} \), we get \( a^{6 - (-3)} = a^{6 + 3} = a^9 \).

So, the simplified numerator is \( a^9 \).

### Step 2: Simplify the Denominator
The denominator is:
\[
a \div \left( a^2 a^3 \right)^4
\]
1. Combine \( a^2 \) and \( a^3 \) inside the parentheses: \( a^2 \cdot a^3 = a^{2+3} = a^5 \).
2. Raise \( a^5 \) to the power of 4: \( (a^5)^4 = a^{5 \cdot 4} = a^{20} \).
3. Now we have \( a \div a^{20} \).
4. Using the rule \( a^m \div a^n = a^{m - n} \), we get \( a^{1 - 20} = a^{-19} \).

So, the simplified denominator is \( a^{-19} \).

### Step 3: Combine Numerator and Denominator
Now we have:
\[
\frac{a^9}{a^{-19}}
\]
Using the division rule \( a^m \div a^n = a^{m - n} \), we get:
\[
a^{9 - (-19)} = a^{9 + 19} = a^{28}
\]

### Final Answer
The expression simplifies to:
\[
a^{28}
\]

Would you like further details or have any questions?

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### Related Questions
1. How do exponent rules work when dealing with negative exponents?
2. What is the rule for distributing exponents over products inside parentheses?
3. How can expressions with fractional exponents be simplified?
4. How would this problem change if we had multiple variables instead of just \( a \)?
5. What is the importance of base assumptions (like \( a \neq 0 \)) in problems involving exponents?

### Tip
Always simplify expressions step-by-step and apply exponent rules carefully to avoid mistakes, especially when dealing with negative exponents.

Represent the following expressions as a power of the number a (a ≠ 0): \[ \frac{\left( a^2 a^4 \right) \div a^{-3}}{a \div \left( a^2 a^3 \right)^4} \]

Learn how to simplify complex expressions with exponents using exponent rules. This problem demonstrates combining, dividing, and raising powers of variable 'a' to simplify the expression as a single power. Suitable for Grades 9-10.

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