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?

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.