Step 1: Understand the question.
We want the ratio of the speed of sound in hydrogen to that in helium at the same temperature. We are given $\gamma_H = \frac{7}{5}$, $\gamma_{He} = \frac{5}{3}$, and molar masses $2$ and $4$.
Step 2: Recall the speed of sound formula.
The speed of sound in a gas is \[ v = \sqrt{\frac{\gamma RT}{M}}, \] where $M$ is the molar mass. Same temperature means $RT$ is common to both.
Step 3: Write the ratio.
\[ \frac{v_H}{v_{He}} = \sqrt{\frac{\gamma_H / M_H}{\gamma_{He} / M_{He}}} = \sqrt{\frac{\gamma_H}{M_H}\cdot\frac{M_{He}}{\gamma_{He}}}. \]
Step 4: Put in the numbers.
\[ \frac{v_H}{v_{He}} = \sqrt{\frac{7/5}{2}\cdot\frac{4}{5/3}} = \sqrt{\frac{7}{10}\cdot\frac{12}{5}}. \]
Step 5: Simplify inside the root.
\[ \frac{7}{10}\cdot\frac{12}{5} = \frac{84}{50} = \frac{42}{25}. \]
Step 6: Take the square root.
\[ \frac{v_H}{v_{He}} = \sqrt{\frac{42}{25}} = \frac{\sqrt{42}}{5}. \] \[ \boxed{\dfrac{v_H}{v_{He}} = \dfrac{\sqrt{42}}{5}} \]