To simplify the given expression $\sqrt{81x^3} \sqrt{81x^5}$, let's break down the steps:

Step-by-step solution:

1. Combine the square roots: $\sqrt{81x^3} \sqrt{81x^5} = \sqrt{(81x^3) \cdot (81x^5)} = \sqrt{81 \cdot 81 \cdot x^3 \cdot x^5}$

2. Simplify the constants and the variables: $81 \cdot 81 = 6561 \quad \text{and} \quad x^3 \cdot x^5 = x^{3+5} = x^8$ So, the expression becomes: $\sqrt{6561 \cdot x^8}$

3. Take the square root:

   • $\sqrt{6561} = 81$, since $81^2 = 6561$.
   • $\sqrt{x^8} = x^4$ because $\sqrt{x^{2n}} = x^n$.

4. Combine the results: $\sqrt{6561 \cdot x^8} = 81x^4$

Final Answer:

$\sqrt{81x^3} \sqrt{81x^5} = 81x^4$

Would you like more details or have any questions about this simplification?

Here are 5 related questions that might interest you:

1. How do you simplify expressions involving square roots and multiplication?
2. What is the rule for combining square roots when multiplying?
3. How do you simplify square roots of variables with exponents?
4. Why does $\sqrt{x^{2n}} = x^n$ only work for positive variables?
5. How would the solution change if the variables weren't assumed to be positive?

Tip: When dealing with square roots of products, it's often helpful to combine them under a single root before simplifying, as it allows easier manipulation of constants and exponents.