The equation given is:

$3^x + \frac{2}{3^{-x}} = 81$

We can start by simplifying the second term, $\frac{2}{3^{-x}}$. This is equivalent to:

$\frac{2}{3^{-x}} = 2 \cdot 3^x$

Thus, the equation becomes:

$3^x + 2 \cdot 3^x = 81$

Now, combine like terms:

$3^x (1 + 2) = 81$

This simplifies to:

$3 \cdot 3^x = 81$

Dividing both sides by 3:

$3^x = \frac{81}{3} = 27$

Since $27 = 3^3$, we can write:

$3^x = 3^3$

Thus, $x = 3$.

The solution is $x = 3$.

Would you like a detailed breakdown of any step, or have any other questions?

Here are 5 related questions:

1. How do you simplify exponential expressions like $\frac{1}{3^{-x}}$?
2. What are the rules for solving equations involving powers with the same base?
3. How would the solution change if the base were different, e.g., 2 instead of 3?
4. How do you graph the function $f(x) = 3^x + \frac{2}{3^{-x}}$?
5. How would this equation change if $81$ were replaced with another number?

Tip: Whenever you see an expression like $\frac{1}{a^{-x}}$, remember that it can be rewritten as $a^x$, which helps simplify calculations!