The problem requires simplifying the expression $\left(\frac{3^{81}}{27^4}\right)^{1/3}$ using exponent rules.

Simplify Base Number

First, express the number 27 as a power of 3:

$27 = 3^3$

Apply Exponent Rules

Substitute $3^3$ for 27 in the expression:

$ \left(\frac{3^{81}}{(3^3)^4}\right)^{1/3} $

Use the power of a power rule, $(a^m)^n = a^{m \times n}$:

$ (3^3)^4 = 3^{3 \times 4} = 3^{12} $

The expression becomes:

$ \left(\frac{3^{81}}{3^{12}}\right)^{1/3} $

Use the quotient rule, $\frac{a^m}{a^n} = a^{m-n}$:

$ \frac{3^{81}}{3^{12}} = 3^{81 - 12} = 3^{69} $

The expression is now simplified to:

$ (3^{69})^{1/3} $

Calculate Final Value

Apply the outer exponent $(1/3)$ using the power of a power rule again:

$ (3^{69})^{1/3} = 3^{69 \times \frac{1}{3}} = 3^{\frac{69}{3}} $

Perform the division:

$ \frac{69}{3} = 23 $

So, the final value is:

$ 3^{23} $

Match with Options

The calculated value $3^{23}$ matches Option C.