${{P}_{1}}=3{{P}_{2}}\Rightarrow {{R}_{1}}I_{1}^{2}=3{{R}_{2}}I_{2}^{2}\Rightarrow 4I_{1}^{2}=9I_{2}^{2}\Rightarrow 2{{I}_{1}}=3{{I}_{2}}$ $\Rightarrow {{I}_{2}}=\frac{2}{3}{{I}_{1}}$    (1) همچنین داریم: ${{I}_{1}}={{I}_{2}}+{{I}_{3}}\xrightarrow{(1)}{{I}_{1}}=\frac{2}{3}{{I}_{1}}+{{I}_{3}}\Rightarrow {{I}_{3}}=\frac{1}{3}{{I}_{1}}$    (2) دو مقاومت ${{R}_{2}}$ و ${{R}_{3}}$ موازی هستند، بنابراین: ${{V}_{2}}={{V}_{3}}\Rightarrow {{R}_{2}}{{I}_{2}}={{R}_{3}}{{I}_{3}}\xrightarrow[(2)]{(1)}3\times \frac{2}{3}{{I}_{1}}={{R}_{3}}\times \frac{1}{3}{{I}_{1}}\Rightarrow {{R}_{3}}=6\Omega $