Let's break down the expression step by step to express it as a single power of $a$.

The given expression is:

$\left( a^2 a^3 \right)^2 \cdot a^4 \div \left( \left( a^{-1} \right)^{-2} \right)^3$

Step 1: Simplify inside each parenthesis

• First, look at $a^2 a^3$. By the law of exponents, $a^m \cdot a^n = a^{m+n}$, so: $a^2 a^3 = a^{2+3} = a^5$
• Now we have $\left( a^5 \right)^2 \cdot a^4 \div \left( \left( a^{-1} \right)^{-2} \right)^3$.

Step 2: Apply exponents

• $\left( a^5 \right)^2$ can be simplified using the power rule $(a^m)^n = a^{m \cdot n}$: $(a^5)^2 = a^{5 \cdot 2} = a^{10}$
• $\left( a^{-1} \right)^{-2}$ simplifies to $a^{(-1)(-2)} = a^2$.

Now the expression becomes: $a^{10} \cdot a^4 \div \left( a^2 \right)^3$

Step 3: Further simplification

• $\left( a^2 \right)^3 = a^{2 \cdot 3} = a^6$, so the expression is now: $a^{10} \cdot a^4 \div a^6$

Step 4: Combine using the rules of exponents

• First, combine $a^{10} \cdot a^4$: $a^{10+4} = a^{14}$
• Now divide $a^{14} \div a^6$ using the division rule $a^m \div a^n = a^{m-n}$: $a^{14-6} = a^8$

Final Answer:

The expression simplifies to: $a^8$

Would you like further clarification or details on any step?

Related Questions:

1. How do the laws of exponents work for multiplication and division?
2. What happens when a negative exponent is raised to another power?
3. How do you handle more complex expressions with different bases?
4. How can you simplify expressions involving both multiplication and division of exponents?
5. Why is the base $a$ required to be non-zero in exponent problems?

Tip: Always apply the exponent rules step-by-step, especially when multiple operations are involved, to avoid errors!