Let \(a\) be the side of the first cube and \(b\) be the diagonal of the second cube and \(v_a\) and \(v_b\) be their volumes respectively.

\(v_a = a^3\)

If x is the side of the second cube then,

\(b^2 = (\sqrt{2} x)^2 +x^2\) \(b^2 = 3x^2\)\(x = \frac{b}{\sqrt{3}}\)

\(v_b = x^3\)

\(v_b = (\frac{b}{\sqrt{3}})^3\)

\(v_b = \frac{b^3}{3^{\frac{3}{2}}}\)

Thus the ratio between the volume of the two cubes is,

\(\frac{v_a}{v_b} = \frac{a^3}{\frac{b^3}{3^{\frac{3}{2}}}}\)

\(\frac{v_a}{v_b} = \frac{a^3}{\frac{a^3}{3^{\frac{3}{2}}}}\) , but \(a = b\)

\(\frac{v_a}{v_b} = \frac{1}{\frac{1}{3^{\frac{3}{2}}}}\)

\(\frac{v_a}{v_b} = 3^{\frac{3}{2}}\)