1. The problem is to rewrite the expression $$\left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^{6}$$ using only positive integer exponents. 2. We use the power of a product rule: $$\left(a^{x} b^{y}\right)^{z} = a^{xz} b^{yz}$$. 3. Apply the rule to each base: $$m^{\frac{2}{3} \times 6} n^{-\frac{1}{3} \times 6} = m^{4} n^{-2}$$. 4. Rewrite the negative exponent as a positive exponent by moving the base to the denominator: $$m^{4} n^{-2} = \frac{m^{4}}{n^{2}}$$. 5. The expression with only positive integer exponents is $$\frac{m^{4}}{n^{2}}$$. 6. Comparing with the options, the correct answer is D.