1. **State the problem:** Simplify the expression $$\sqrt{\sqrt{\left(\frac{m^{3}}{m^{-11}}\right)^{4}}}$$. 2. **Use exponent rules:** Recall that $$\frac{a^{x}}{a^{y}} = a^{x-y}$$ and $$\left(a^{b}\right)^{c} = a^{bc}$$. 3. **Simplify inside the parentheses:** $$\frac{m^{3}}{m^{-11}} = m^{3 - (-11)} = m^{3 + 11} = m^{14}$$ 4. **Raise to the 4th power:** $$\left(m^{14}\right)^{4} = m^{14 \times 4} = m^{56}$$ 5. **Apply the nested square roots:** $$\sqrt{\sqrt{m^{56}}} = \sqrt{m^{\frac{56}{2}}} = \sqrt{m^{28}}$$ 6. **Simplify the outer square root:** $$\sqrt{m^{28}} = m^{\frac{28}{2}} = m^{14}$$ **Final answer:** $$m^{14}$$