We need to solve the equation $3^k = \frac{81^2 \times 3^5}{3^{11}}$ for the value of $k$.

1. Problem Description

We need to solve the equation $3^k = \frac{81^2 \times 3^5}{3^{11}}$ for the value of $k$.

2. Solution Steps

First, we rewrite $81$ as $3^4$. Then the equation becomes:

$3^k = \frac{(3^4)^2 \times 3^5}{3^{11}}$

Next, we simplify $(3^4)^2$ using the power of a power rule $(a^m)^n = a^{m \times n}$.

$(3^4)^2 = 3^{4 \times 2} = 3^8$

Substitute this back into the equation:

$3^k = \frac{3^8 \times 3^5}{3^{11}}$

Now, we simplify the numerator using the rule $a^m \times a^n = a^{m+n}$:

$3^8 \times 3^5 = 3^{8+5} = 3^{13}$

Substitute this back into the equation:

$3^k = \frac{3^{13}}{3^{11}}$

Next, we simplify the fraction using the rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{3^{13}}{3^{11}} = 3^{13-11} = 3^2$

So the equation becomes:

$3^k = 3^2$

Since the bases are equal, the exponents must be equal. Therefore,

$k = 2$

3. Final Answer

The final answer is $k = 2$.