Let's solve the given equation:

$3^{2a} \times 81^b = 3^n$

Step 1: Express $81^b$ in terms of base 3.

Since $81 = 3^4$, we can rewrite $81^b$ as:

$81^b = (3^4)^b = 3^{4b}$

Step 2: Substitute into the original equation.

The equation now becomes:

$3^{2a} \times 3^{4b} = 3^n$

Step 3: Apply the property of exponents.

Using the rule $x^m \times x^n = x^{m+n}$, combine the terms on the left-hand side:

$3^{2a + 4b} = 3^n$

Step 4: Equate the exponents.

Since the bases are the same, we can set the exponents equal to each other:

$2a + 4b = n$

Final Answer:

Thus, $n$ in terms of $a$ and $b$ is:

$n = 2a + 4b$

Would you like further details or have any questions?

Related Questions:

1. How would the solution change if $81$ was replaced by another power of 3?
2. What if the original equation had another base instead of 3?
3. Can you solve for $a$ in terms of $n$ and $b$?
4. How would the solution be affected if the exponents were fractional?
5. How can this technique be applied to solve exponential equations with other bases?

Tip:

When solving exponential equations, always try to express all terms with the same base to simplify the equation.