We have $$\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b}=\gamma a^{1/n} + (1-\gamma)b^{1/n} \\ =\gamma\Big(1+\frac1n\,\ln a+O(1/n^2)\Big) + (1-\gamma)\Big(1+\frac1n\,\ln b+O(1/n^2)\Big) \\ =1+\frac1n\,\ln(a^\gamma b^{1-\gamma})+O(1/n^2)\\ =\exp\Big\{\frac1n\,\ln(a^\gamma b^{1-\gamma})+O(1/n^2)\Big\},$$ whence the limit is $a^\gamma b^{1-\gamma}$.

We have $$\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b}=\gamma a^{1/n} + (1-\gamma)b^{1/n} \\ =\gamma e^{(1/n)\ln a} + (1-\gamma)e^{(1/n)\ln b} \\ =\gamma\Big(1+\frac1n\,\ln a+O(1/n^2)\Big) + (1-\gamma)\Big(1+\frac1n\,\ln b+O(1/n^2)\Big) \\ =1+\frac1n\,\ln(a^\gamma b^{1-\gamma})+O(1/n^2)\\ =\exp\Big\{\frac1n\,\ln(a^\gamma b^{1-\gamma})+O(1/n^2)\Big\},$$ whence the limit is $a^\gamma b^{1-\gamma}$.