Simplify an algebraic expression with exponents.

1. Apply the exponent rule $(a^m)^n = a^{m \cdot n}$:
We need to distribute the outer exponent of 2 to each factor inside the parentheses. The expression is $(11c^3 d^2 g^2)^2$. Applying the rule, we get:
$11^2 \cdot (c^3)^2 \cdot (d^2)^2 \cdot (g^2)^2$

2. Simplify each term:
- For the constant term: $11^2 = 11 \times 11 = 121$
- For the variable $c$: $(c^3)^2 = c^{3 \times 2} = c^6$
- For the variable $d$: $(d^2)^2 = d^{2 \times 2} = d^4$
- For the variable $g$: $(g^2)^2 = g^{2 \times 2} = g^2$

3. Combine the simplified terms:
Putting all the simplified terms back together, we get:
$121 c^6 d^4 g^2$

4. Express the answer using exponents as requested:
The problem asks to express the answer using exponents. While $121$ is a simplified constant, it can also be expressed as $11^2$. Therefore, the final answer in the requested format is:
$11^2 c^6 d^4 g^2$