Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*} a^2/b &= 1, \\ b^2/c &= 2, \text{ and}\\ c^2/a &= 3. \end{align*} Find $a$.
+26367 Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*} a^2/b &= 1, \\ b^2/c &= 2, \text{ and}\\ c^2/a &= 3. \end{align*} Find $a$.
\(\begin{array}{|rcll|} \hline (1) & a^2 &=& b \\ & a &=& \sqrt{b} \\\\ (2) & b^2 &=& 2c \\ & b &=& \sqrt{2c} \\\\ (3) & c^2 &=& 3a \quad & | \quad a = \sqrt{b}\\ & c^2 &=& 3 \sqrt{b} \quad & | \quad b = \sqrt{2c}\\ & c^2 &=& 3 \sqrt{\sqrt{2c}} \\ & c^2 &=& 3 \sqrt[4]{ 2c } \\ & (c^2)^4 &=& 3^4 2c \\ & c^8 &=& 162c \\ & c^7 &=& 162 \\ & c &=& \sqrt[7]{ 162 } \\ & \mathbf{c} & \mathbf{=} & \mathbf{2.06844987991} \\\\ & a &=& \frac{c^2}{3} \\ & a &=& \frac{2.06844987991^2}{3} \\ & a &=& \frac{4.27848490568}{3} \\ & \mathbf{a} & \mathbf{=} & \mathbf{1.42616163523} \\\\ & b &=& a^2 \\ & b &=& 1.42616163523^2 \\ & \mathbf{b} & \mathbf{=} & \mathbf{2.03393700979} \\ \hline \end{array}\)
