Answer
$\frac{c^*}{c_0}=\left(\frac{2}{\boldsymbol{k}+\mathbf{1}}\right)^{1 / 2}
$
Work Step by Step
For an ideal gas the speed of sound is expressed as $c=\sqrt{k R T}$. Thus,
$$
\frac{c^*}{c_0}=\frac{\sqrt{k R T^*}}{\sqrt{k R T_0}}=\left(\frac{T^*}{T_0}\right)^{1 / 2}\\=\left(\frac{2}{\boldsymbol{k}+\mathbf{1}}\right)^{1 / 2}
$$
