ABC26GN1414 · Square Roots and Cube Roots

$$\begin{aligned} & a+b+c=27 \sqrt{29} \Rightarrow 2 c+\frac{3}{2} c+c=27 \sqrt{29} \\ & \Rightarrow \frac{9}{2} c=27 \sqrt{29} \Rightarrow c=6 \sqrt{29} \\ & \sqrt{a^{2}+b^{2}+c^{2}}=\sqrt{(a+b+c)^{2}-2(a b+b c+c a)} \\ &=\sqrt{(27 \sqrt{29})^{2}-2\left(2 c \times \frac{3}{2} c+\frac{3}{2} c \times c+c \times 2 c\right)} \\ &=\sqrt{(729 \times 29)-2\left(3 c^{2}+\frac{3}{2} c^{2}+2 c^{2}\right)} \\ &=\sqrt{(729 \times 29)-2 \times \frac{13}{2} c^{2}} \\ &=\sqrt{(729 \times 29)-13 \times(6 \sqrt{29})^{2}}=\sqrt{29(729-468)} \\ &=\sqrt{29 \times 261}=\sqrt{29 \times 29 \times 9}=29 \times 3=87 \end{aligned}$$